Expansions of the Solutions of the General Heun …downloads.hindawi.com/journals/ahep/2018/4263678.pdfExpansions of the Solutions of the General Heun Equation Governed by Two-Term

Research ArticleExpansions of the Solutions of the General Heun EquationGoverned by Two-Term Recurrence Relations for Coefficients

T A Ishkhanyan123 T A Shahverdyan3 and A M Ishkhanyan 14

1Russian-Armenian University H Emin 123 0051 Yerevan Armenia2Moscow Institute of Physics and Technology Dolgoprudny Moscow Region 141700 Russia3Institute for Physical Research NAS of Armenia Ashtarak 0203 Armenia4Institute of Physics and Technology National Research Tomsk Polytechnic University Tomsk 634050 Russia

Correspondence should be addressed to A M Ishkhanyan aishkhanyangmailcom

Received 27 January 2018 Accepted 4 May 2018 Published 4 July 2018

Academic Editor Saber Zarrinkamar

Copyright copy 2018 TA Ishkhanyan et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

We examine the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions Wepresent several expansions using functions the forms of which differ from those applied before In general the coefficients of theexpansions obey three-term recurrence relations However there exist certain choices of the parameters for which the recurrencerelations become two-term The coefficients of the expansions are then explicitly expressed in terms of the gamma functionsDiscussing the termination of the presented series we show that the finite-sum solutions of the general Heun equation in termsof generally irreducible hypergeometric functions have a representation through a single generalized hypergeometric functionConsequently the power-series expansion of the Heun function for any such case is governed by a two-term recurrence relation

1 Introduction

The general Heun equation [1ndash3] which is the most generalsecond-order linear ordinary differential equation havingfour regular singular points is currently widely encounteredin physics andmathematics research (see eg [1ndash14] and ref-erences therein) However this equation is much less studiedthan its immediate predecessor the Gauss hypergeometricequation which is the most general equation having threeregular singular points A reason for the slow progress in thedevelopment of the theory is that the solutions of the Heunequation (as well as its four confluent reductions) in generalare not expressed in terms of definite or contour integralsinvolving simpler functions [2 3] Furthermore the con-vergence regions of power-series expansions near differentsingularities are rather restricted and several complicationsarise in studying the relevant connection problems [2 315] Another general problematic point is that the power-series solutions of the Heun equation are governed by three-term recurrence relations between successive coefficients ofexpansions [1ndash3] instead of two-term ones appearing in

the hypergeometric case [16ndash18] As a result in general thecoefficients are not determined explicitly

In the present paper we show that there exist someparticular choices of the involved parameters for which therecurrence relations governing the power-series expansionsbecome two-term In these cases the solution of the generalHeun equation can be written either as a linear combinationof a finite number of the Gauss hypergeometric functions orin terms of a single generalized hypergeometric functionThisis a main result of the present paper

Another major result we report here is that in the caseof the expansions of the solutions of the Heun equation interms of hypergeometric functions there also exist particularchoices of the involved parameters for which the governingthree-term recurrence relations for expansion coefficientsbecome two-term In these cases the coefficients are explicitlywritten in terms of the gamma functions

Expansions of the solutions of the Heun equation interms of the Gauss hypergeometric functions 21198651 initiatedby Svartholm [19] suggest a notable extension of the seriestechnique This is a useful approach applicable to many

HindawiAdvances in High Energy PhysicsVolume 2018 Article ID 4263678 9 pageshttpsdoiorg10115520184263678

2 Advances in High Energy Physics

differential equations including those of more general typewhose nature outside a certain region of the extendedcomplex plane containing only two regular singular pointsis not necessary to be specified exactly Expansions involvingfunctions other than powers have been applied to the generaland confluentHeun equations bymany authorsThe ordinaryhypergeometric [19ndash25] confluent hypergeometric [26ndash29]Coulomb wave functions [30 31] Bessel and Hankel func-tions [32] incomplete Beta and Gamma functions [33ndash35]Hermite functions [36 37] Goursat and Appell generalizedhypergeometric functions of two variables of the first kind[38 39] and other known special functions have been usedas expansion functions A useful property suggested by theseexpansions is the possibility of deriving finite-sum solutionsby means of termination of the series

As far as the expansions of the general Heun equation interms of the hypergeometric functions are concerned in theearly papers by Svartholm [19] Erdelyi [20 21] and Schmidt[22] the intuitive intention was to apply hypergeometricfunctions with parameters so chosen as to match the Heunequation as closely as possible For this reason they used func-tions of the form 21198651(120582 + 119899 120583 minus 119899 120574 119911) which have matchingbehavior in two singular points 119911 = 0 and 1 These functionshave the following Riemann 119875-symbol representation

119875( 0 1 infin0 0 120582 + 119899 1199111 minus 120574 1 minus 120575 120583 minus 119899 ) (1)

Here 120582 and 120583may adopt several values provided 1 + 120582 + 120583 =120574 + 120575 (see [25]) It is clear that the functions have matchingcharacteristic exponents at 119911 = 0 and 119911 = 1 and theirbehavior does not match that of the Heun function at thethird singular point of the hypergeometric equation 119911 = infin

However it has been shown that one can also usefunctions that have matching behavior at only one singularpoint [38] Exploring this idea in the present paperwe discussthe hypergeometric expansions of the solutions of the Heunequation in terms of functions of the form 21198651(120572 120573 1205740 plusmn119899 119911)which havematching behavior (ie characteristic exponents)only at the singular point 119911 = infin These functions arepresented by the Riemann 119875-symbol

119875( 0 1 infin0 0 120572 1199111 minus (1205740 plusmn 119899) 1 minus (1205750 ∓ 119899) 120573 ) (2)

where 1 + 120572 + 120573 = 1205740 + 1205750 and the parameters 1205740 1205750 arechosen so that the Fuchsian condition for the general Heunequation (3) ie 1 + 120572 + 120573 = 120574 + 120575 + 120576 is fulfilled 1205740 + 1205750 =120574+120575+120576 Note that these functions differ also from the Jacobi-polynomials used by Kalnins andMiller whose functions canbe written in terms of hypergeometric functions of the form21198651(120582 + 119899 120583 minus 119899 ] + 2119899 119911) [23]

In the present paper we discuss several expansions interms of thementioned hypergeometric functions In generalthe coefficients of the expansions obey three-term recurrencerelations similar to those known fromprevious developments

[19ndash25] However for certain choices of the involved param-eters the recurrence relations reduce to two-term ones Inthese exceptional cases the coefficients of the expansions areexplicitly calculated The result is expressed in terms of thegamma functions

Discussing the conditions for deriving finite-sum solu-tions by means of termination of the presented series weshow that the termination is possible if a singularity of theHeun equation is an apparent one Furthermore we show thatany finite-sum solution of the general Heun equation derivedin this way has a representation through a single generalizedhypergeometric function 119901119865119902 The general conclusion is thenthat in any such case the power-series expansion of theHeun function is governed by a two-term recurrence relation(obviously this is the relation obeyed by the correspondingpower-series for 119901119865119902)2 Hypergeometric Expansions

Thegeneral Heun equation written in its canonical form is [1]

11988921199061198891199112 + (120574119911 + 120575119911 minus 1 + 120576119911 minus 119886) 119889119906119889119911 + 120572120573119911 minus 119902119911 (119911 minus 1) (119911 minus 119886)119906= 0 (3)

where the parameters satisfy the Fuchsian relation 1+120572+120573 =120574+120575+120576We introduce an expansion of this equationrsquos solutionof the form

119906 = infinsum119899=0

119888119899 sdot 21198651 (120572 120573 1205740 + 119899 119911) (4)

with the involved Gauss hypergeometric functions obeyingthe equation

11988921199061198991198891199112 + (1205740 + 119899119911 + 120575 + 120576 + 120574 minus 1205740 minus 119899119911 minus 1 ) 119889119906119899119889119911+ 120572120573119911 (119911 minus 1)119906119899 = 0

(5)

Substitution of (4)-(5) into (3) gives

sum119899

119888119899 [(120574 minus 1205740 minus 119899119911 minus 120576 + 120574 minus 1205740 minus 119899119911 minus 1 + 120576119911 minus 119886) 119889119906119899119889119911+ 120572120573119886 minus 119902119911 (119911 minus 1) (119911 minus 119886)119906119899] = 0

(6)

or sum119899

119888119899 [(119886 minus 1) (120576 + 120574 minus 1205740 minus 119899) 119911minus 119886 (120574 minus 1205740 minus 119899) (119911 minus 1)] 119889119906119899119889119911 + (120572120573119886 minus 119902)sdot 119906119899 = 0

(7)

Advances in High Energy Physics 3

Now using the following relations between the involvedhypergeometric functions [16ndash18]

119911119889119906119899119889119911 = 120574119899minus1 [119906119899minus1 minus 119906119899] (8)

(119911 minus 1) 119889119906119899119889119911 = minus120575119899+1119906119899 + (120575119899+1 minus 120572120573120574119899 )119906119899+1 (9)

where 120574119899 = 1205740 + 119899 and 120575119899 = 120575 + 120576 + 120574 minus 120574119899 this equation isrewritten as

sum119899

119888119899 [(119886 minus 1) (120576 + 120574 minus 120574119899) (120574119899 minus 1) [119906119899minus1 minus 119906119899]+ 119886 (120574 minus 120574119899) ((120575119899 minus 1) 119906119899 minus (120575119899 minus 1 minus 120572120573120574119899 )119906119899+1)+ (120572120573119886 minus 119902) 119906119899] = 0

(10)

from which we get a three-term recurrence relation for thecoefficients of the expansion (4)

119877119899119888119899 + 119876119899minus1119888119899minus1 + 119875119899minus2119888119899minus2 = 0 (11)

with

119877119899 = (119886 minus 1) (120576 + 120574 minus 120574119899) (120574119899 minus 1) (12)

119876119899 = minus119877119899 + 119886 (120574 minus 120574119899) (120572 + 120573 minus 120574119899) + (120572120573119886 minus 119902) (13)

119875119899 = 119886120574119899 (120574 minus 120574119899) (120572 minus 120574119899) (120573 minus 120574119899) (14)

From the initial conditions 1198880 = 1 and 119888minus1 = 119888minus2 = 0we get (120576+120574minus1205740)(1205740minus1) = 0 Since 1205740 = 1 is forbidden (it causes divisionby zero at 119899 = 1 in 119875minus1) we obtain that the only possibility is1205740 = 120576 + 120574 Hence the expansion finally reads

119906 = infinsum119899=0

119888119899 sdot 21198651 (120572 120573 120574 + 120576 + 119899 119911) (15)

and the coefficients of the three-term recurrence relation (11)are

119877119899 = (1 minus 119886) 119899 (120576 + 120574 + 119899 minus 1) (16)

119876119899 = minus119877119899 + 119886 (1 + 119899 minus 120575) (119899 + 120576) + (119886120572120573 minus 119902) (17)

119875119899= minus 119886119899 + 120576 + 120574 (119899 + 120576) (119899 + 120576 + 120574 minus 120572) (119899 + 120576 + 120574 minus 120573) (18)

The expansion applies if 120572 120573 and 120574 + 120576 are not zero ornegative integers The restrictions on 120572 and 120573 assure thatthe hypergeometric functions are not polynomials of fixeddegree

The derived expansion terminates if two successive coef-ficients vanish If 119888119873 is the last nonzero coefficient and 119888119873+1 =

119888119873+2 = 0 for some119873 = 0 1 2 we obtain from (11) that itshould be 119875119873 = 0 so that the termination is possible if

120576 = minus119873or 120576 + 120574 minus 120572 = minus119873or 120576 + 120574 minus 120573 = minus119873 (19)

Note that the equation 119888119873+1 = 0 results in a polynomialequation of degree119873 + 1 for the accessory parameter 119902 Thisequation is convenient for rewriting the recurrence relation(11) in the following matrix form

[[[[[[[[[

1198760 1198771 01198750 1198761 1198772 00 1198751 1198762 1198773d d d0 119875119873minus1 119876119873

]]]]]]]]]

[[[[[[[[[[

119888011988811198882119888119873

]]]]]]]]]]=[[[[[[[[[[

0000

]]]]]]]]]] (20)

The vanishing of the determinant of the above matrix givesthe polynomial equation for 119902 defining in general119873+1 valuesfor which the termination occurs

One may consider a mirror expansion

119906 = infinsum119899=0

119888119899 sdot 21198651 (120572 120573 1205740 minus 119899 119911) (21)

which differs from expansion (4) only by the sign of 119899 in thelower parameter of the involved hypergeometric functionsThis change of the sign leads to a three-term recurrencerelation (11) with the coefficients

119877119899 = 1198861205740 minus 119899 (120574 minus 1205740 + 119899) (120572 minus 1205740 + 119899) (120573 minus 1205740 + 119899) (22)

119876119899 = minus119875119899 + 119886 (120574 minus 1205740 + 119899) (120572 + 120573 minus 1205740 + 119899) + 120572120573119886minus 119902 (23)

119875119899 = (119886 minus 1) (120576 + 120574 minus 1205740 + 119899) (1205740 minus 119899 minus 1) (24)

where

1205740 = 120574 or 120572 or 120573 (25)

This expansion applies if 120572 and 120573 are not zero or negativeintegers and 120574 is not an integer

In order for the series to terminate at some 119899 = 119873 we put119875119873 = 0 so that this time we derive

120576 120576 + 120574 minus 120572or 120576 + 120574 minus 120573 = minus119873 (26)

for the expansions with 1205740 = 120574 or 120572 or 120573 respectively Thenthe equation 119888119873+1 = 0 again gives a (119873+1)-degree polynomialequation for those values of the accessory parameter 119902 forwhich the termination occurs


3 Finite-Sum Hypergeometric Solutions

It is readily shown that the finite-sum solutions derivedfrom the above two types of expansions by the describedtermination procedure coincide as can be expected becauseof apparent symmetry For example consider the expansion(15) in the case 120576 = minus119873 The involved hypergeometricfunctions have the form 119906119899 = 21198651(120572 120573 120574minus119873+119899 119911) Since 119899 =0 1 119873 we see that the set of the involved hypergeometricfunctions is exactly the same as in the case of the second typeexpansion (21) with 1205740 = 120574 21198651(120572 120573 120574 119911) 21198651(120572 120573 120574 minus1 119911) 21198651(120572 120573 120574 minus 119873 119911) Furthermore examination of(20) shows that the equation for the accessory parameter 119902and the expansion coefficients are also the same for the twoexpansionsThe same happens to other two cases 120576 + 120574 minus 120572 =minus119873 and 120576 + 120574 minus 120573 = minus119873 Thus while different in general theexpansions (15)-(18) and (21)-(25) lead to the same finite-sumclosed-form solutions

Consider the explicit forms of these solutions examin-ing for definiteness the expansion (15)-(18) An immediateobservation is that because of the symmetry of the Heunequation with respect to the interchange 120572 larrrarr 120573 the finite-sum solutions produced by the choices 120576 + 120574 minus 120572 = minus119873 and120576+120574minus120573 = minus119873 are of the same form Furthermore by applyingthe formula [16]

21198651 (120572 120573 120572 + 119896 119911)= (1 minus 119911)119896minus120573 21198651 (119896 120572 minus 120573 + 119896 120572 + 119896 119911) (27)

to the involved hypergeometric functions 21198651(120572 120573 120572 minus 119873 +119899 119911) or 21198651(120572 120573 120573 minus119873 + 119899 119911) we see that the sum is a quasi-polynomial namely a product of (1minus119911)1minus120575 and a polynomialin 119911 Here are the first two of the solutions120576 + 120574 minus 120572 = 0119902 = 119886120574 (120575 minus 1) 119906 = (1 minus 119911)1minus120575

(28)

120576 + 120574 minus 120572 = minus11199022 + [120572 minus 1 minus 119886 (120575 minus 2 + 120574 (2120575 minus 3))] 119902 minus 119886120574 (120575 minus 2)

sdot (120572 minus 119886 (1 + 120574) (120575 minus 1)) = 0(29)

119906 = (1 minus 119911)1minus120575sdot (1 minus 120572 + 1 minus 120575120572 minus 1 119911 + 119902 minus 119886 (120572120573 + 120576 minus 120575120576)(1 minus 119886) (120572 minus 1) (1 minus 119911)) (30)

Note that since 1 minus 120575 is a characteristic exponent of theHeun equation the transformation 119906 = (1minus119911)1minus120575119908(119911) resultsin another Heun equation for119908(119911) Hence the derived finite-sum solutions corresponding to the choices 120576+120574minus120572 = minus119873 and120576 + 120574 minus 120573 = minus119873 are generated from the polynomial solutionsof the equation for 119908(119911)

More interesting is the case 120576 = minus119873 when the finite-sumsolutions involve119873+1 hypergeometric functions irreduciblein general to simpler functions The case 120576 = 0 produces the

trivial result 119902 = 119886120572120573 when theHeun equation is degeneratedinto the hypergeometric equation with the solution 119906 =21198651(120572 120573 120574 119911)The solution for the first nontrivial case 120576 = minus1reads

119906 = 21198651 (120572 120573 120574 minus 1 119911) + 119902 minus 119886120572120573 + 119886 (1 minus 120575)(1 minus 119886) (120574 minus 1)sdot 21198651 (120572 120573 120574 119911) (31)

where 119902 is a root of the equation(119902 minus 119886120572120573 + 119886 (1 minus 120575)) (119902 minus 119886120572120573 + (119886 minus 1) (1 minus 120574))

minus 119886 (1 minus 119886) (1 + 120572 minus 120574) (1 + 120573 minus 120574) = 0 (32)

Note that the second term in (31) vanishes if 119902 minus 119886120572120573 + 119886(1 minus120575) = 0 so that in this degenerate case the solution involvesone not 2 = 119873 + 1 terms It is seen from (32) that thissituation is necessarily the case if 119886 = 12 120574 + 120575 = 2 and120572 or 120573 equals 120574 minus 1 We will see that the solution in this caseis a member of a family of specific solutions for which theexpansion is governed by two-term recurrence relations forthe coefficients

The solutions (31) and (32) have been noticed on severaloccasions [40ndash44] It has been shown that for 120576 = minus1when the characteristic exponents of 119911 = 119886 are 0 2 sothat they differ by an integer (32) provides the conditionfor the singularity 119911 = 119886 to be apparent (or ldquosimplerdquo) thatis no logarithmic terms are involved in the local Frobeniusseries expansion [40ndash42] In fact the Frobenius solution inthis case degenerates to a Taylor series It has further beenobserved that the solution (31) can be expressed in terms oftheClausen generalized hypergeometric function 31198652with anupper parameter exceeding a lower one by unity [40ndash42]

119906119906 (0) = 31198652 (120572 120573 119890 + 1 120574 119890 119911) (33)

where the parameter 119890 is given as

119890 = 119886120572120573119902 minus 119886120572120573 (34)

Note that using this parameter 119890 the solution of (32) isparameterized as [41]

119902 = 1198861205721205731 + 119890119890 119886 = 119890 (119890 minus 120574 + 1)(119890 minus 120572) (119890 minus 120573)

(35)

We will now show that a similar generalized hypergeometricrepresentation holds also for 120576 = minus2 and for all 120576 isin Z 120576 = 1


4 The Case 120576 le minus2 120576 isin ZFor 120576 = minus2 the termination equation 119888119873+1 = 0 for theaccessory parameter 119902 is written as

((119902 minus 119886120572120573)2 + (119902 minus 119886120572120573) (4119886 minus 2 minus (3 + 120572 + 120573) 119886 + 120574)+ 2119886 (119886 minus 1) 120572120573) times (119902 minus 119886120572120573 minus 2 (1 + 120572 + 120573) 119886 minus 2+ 2120574) + (119902 minus 119886120572120573) 2119886 (119886 minus 1) (120572120573 + 1 + 120572 + 120573)= 0

(36)

The solution of the Heun equation for a root of this equationis given as

119906 = 21198651 (120572 120573 120574 minus 2 119911) + 1198611 sdot 21198651 (120572 120573 120574 minus 1 119911) + 1198612sdot 21198651 (120572 120573 120574 119911) (37)

with

1198611 = 119902 minus 119886120572120573 + 119886 (1 minus 120575)(1 minus 119886) (120574 minus 2) 1198612 = 119886 (1 + 120572 minus 120574) (1 + 120573 minus 120574)(119902 minus 119886120572120573 + 2 (1 minus 119886) (120574 minus 1)) (120574 minus 1)1198611

(38)

It is now checked that this solution is presented by thehypergeometric function 41198653 as [42]

119906119906 (0) = 41198653 (120572 120573 119890 + 1 119903 + 1 120574 119890 119903 119911) (39)

where 119906(0) = 1 + 1198611 + 1198612 and the parameters 119890 119903 solve theequations (compare with (35))

119902 = 119886120572120573(1 + 119890) (1 + 119903)119890119903 (40)

119886 = 119890119903 (2119890119903 + (119890 + 119903 + 1) (2 minus 120574))120572120573 ((1 + 119890)2 + (1 + 119903)2 minus 1) + 119890119903 (2119890119903 minus 4 minus (119890 + 119903 + 3) (120572 + 120573 minus 1)) (41)

It is further checked that this system of equations admits aunique solution 119890 119903 (up to the transposition 119890 larrrarr 119903)

The presented result is derived in a simple way bysubstituting the ansatz (39) into the general Heun equationand expanding the result in powers of 119911 The equationsresulting in cancelling the first three terms proportional to 11991101199111 and 1199112 are that given by (40) (41) and (36) respectivelyIt is then shown that these three equations are enough for theHeun equation to be satisfied identically

A further remark is that (36) presents the condition forthe singularity 119911 = 119886 to be apparent for 120576 = minus2 Thisis straightforwardly verified by checking the power-seriessolution 119906 = suminfin119899=0 119888119899(119911minus119886)119899 with 1198880 = 0 for the neighborhoodof the point 119911 = 119886 In calculating 1198883 a division by zero willoccur unless 119902 satisfies (36) in which case the equation for 1198883will be identically satisfied

It can be checked that generalized hypergeometric rep-resentations are achieved also for 120576 = minus3 minus4 minus5 [42] Theconjecture is that for any negative integer 120576 = minus119873 119873 =1 2 3 there exists a generalized hypergeometric solutionof the Heun equation given by the ansatz119906

= 119873+21198651+119873 (120572 120573 1198901 + 1 119890119873 + 1 120574 1198901 119890119873 119911) (42)

provided the singularity at 119911 = 119886 is an apparent one Forthe latter condition to be the case the accessory parameter119902 should satisfy a (119873 + 1)-degree polynomial equationwhich forces the above expansions (15)-(18) and (21)-(25) toterminate at119873th term Note that (42) applies also for119873 = 0that is for 120576 = 0 for which the Heun function degenerates

to the Gauss hypergeometric function 119906 = 21198651(120572 120573 120574 119911)provided 119902 = 0

Finally we note that by the elementary power change119906 = (119911 minus 119886)1minus120576119908 a Heun equation with a positive exponentparameter 120576 gt 2 is transformed into the one with a negativeparameter 2 minus 120576 lt 0 Hence it is understood that a similargeneralized hypergeometric representation of the solution oftheHeun equation can also be constructed for positive integer120576 = 119873 119873 = 2 3 Thus the only exception is the case120576 = 1

It is a basic knowledge that the generalized hypergeomet-ric function 119901119865119902 is given by a power-series with coefficientsobeying a two-term recurrence relation Since any finite-sumsolution of the general Heun equation derived via termina-tion of a hypergeometric series expansion has a representa-tion through a single generalized hypergeometric function119901119865119902 the general conclusion is that in each such case thepower-series expansion of the Heun function is governedby a two-term recurrence relation (obviously by the relationobeyed by the corresponding power-series for 119901119865119902)5 Hypergeometric Expansions with Two-TermRecurrence Relations for the Coefficients

In this section we explore if the three-term recurrence rela-tions governing the above-presented hypergeometric expan-sions can be reduced to two-term ones We will see that theanswer is positive Two-term reductions are achieved for aninfinite set of particular choices of the involved parameters

First we mention a straightforward case which actuallyturns to be rather simple because in this case the Heun


equation is transformed into the Gauss hypergeometricequation by a variable change This is the case if119886 = 12120574 + 120575 = 2

and 119902 = 119886120572120573 + 119886 (1 minus 120575) 120576 (43)

when the coefficient 119876119899 in (11) identically vanishes so thatthe recurrence relation between the expansion coefficientsstraightforwardly becomes two-term for both expansions(15)-(18) and (21)-(25) The coefficients of the expansionsare then explicitly calculated For instance expansion (15) iswritten as

119906 = infinsum119896=0

(1205762)119896 ((120574 + 120576 minus 120572) 2)119896 ((120574 + 120576 minus 120573) 2)119896119896 ((120574 + 120576) 2)119896 ((1 + 120574 + 120576) 2)119896sdot 21198651 (120572 120573 120574 + 120576 + 2119896 119911) (44)

where ( )119896 denotes the Pochhammer symbol The values119906(0) 1199061015840(0) and 119906(1) and 1199061015840(1) can then be written in termsof generalized hypergeometric series [17 18] For instance

119906 (0) = 31198652 (120574 + 120576 minus 1205722 120574 + 120576 minus 1205732 1205762 120574 + 1205762 1 + 120574 + 1205762 1) (45)

However as it was alreadymentioned above the case (43)is a rather simple one because the transformation

119906 (119911) = 1199111minus120574 (1 minus 119911119886)1minus120576 119908 (4119911 (1 minus 119911)) (46)

reduces the Heun equation to the Gauss hypergeometricequation for the new function 119908 The solution of the generalHeun equation is then explicitly written as

119906 (119911) = 1199111minus120574 (1 minus 119911119886)1minus120576sdot 21198651 (1 minus 120572 + 1205752 1 minus 120573 + 1205752 120575 4 (1 minus 119911) 119911) (47)

Now we will show that there exist nontrivial cases oftwo-term reductions of the three-term recurrence (11) with(16)-(18) or (22)-(24) These reductions are achieved by thefollowing ansatz guessed by examination of the structure ofsolutions (33) and (39)

119888119899 = (1119899 prod119873+2119896=1 (119886119896 minus 1 + 119899)prod119873+1119896=1 (119887119896 minus 1 + 119899)) 119888119899minus1 (48)

where having in mind the coefficients 119877119899 119876119899 and 119875119899 givenby (16)-(18) we put1198861 119886119873 119886119873+1 119886119873+2= 1 + 1198901 1 + 119890119873 120574 + 120576 minus 120572 120574 + 120576 minus 120573 (49)

1198871 119887119873 119887119873+1 = 1198901 119890119873 120574 + 120576 (50)

with parameters 1198901 119890119873 to be defined later Note that thisansatz implies that 1198901 119890119873 are not zero or negative integers

The ratio 119888119899119888119899minus1 is explicitly written as

119888119899119888119899minus1 = (120574 + 120576 minus 120572 minus 1 + 119899) (120574 + 120576 minus 120572 minus 1 + 119899)(120574 + 120576 minus 1 + 119899) 119899 119873prod119896=1

sdot 119890119896 + 119899119890119896 minus 1 + 119899 (51)

With this the recurrence relation (11) is rewritten as

119877119899 (120574 + 120576 minus 120572 minus 1 + 119899) (120574 + 120576 minus 120573 minus 1 + 119899)(120574 + 120576 minus 1 + 119899) 119899 119873prod119896=1

sdot 119890119896 + 119899119890119896 minus 1 + 119899 + 119876119899minus1 + 119875119899minus2sdot (120574 + 120576 minus 2 + 119899) (119899 minus 1)(120574 + 120576 minus 120572 minus 2 + 119899) (120574 + 120576 minus 120573 minus 2 + 119899) 119873prod

119896=1

sdot 119890119896 minus 2 + 119899119890119896 minus 1 + 119899 = 0

(52)

Substituting119877119899 and119875119899minus2 from (16) and (18) and cancelling thecommon denominator this equation becomes

(1 minus 119886) (120574 + 120576 minus 120572 minus 1 + 119899) (120574 + 120576 minus 120573 minus 1 + 119899)sdot 119873prod119896=1

(119890119896 + 119899) + 119876119899minus1 119873prod119896=1

(119890119896 minus 1 + 119899)minus 119886 (120576 + 119899 minus 2) (119899 minus 1) 119873prod

119896=1

(119890119896 minus 2 + 119899) = 0(53)

This is a polynomial equation in 119899 Notably it is of degree119873+1 not119873+2 because the highest-degree term proportional to119899119873+2 identically vanishes Hence we have an equation of theform

119873+1sum119898=0

119860119898 (119886 119902 120572 120573 120574 120575 120576 1198901 119890119873) 119899119898 = 0 (54)

Then equating to zero the coefficients 119860119898 warrants thesatisfaction of the three-term recurrence relation (11) for all 119899We thus have119873+2 equations119860119898 = 0119898 = 0 1 119873+1 ofwhich119873 equations serve for determination of the parameters11989012119873 and the remaining two impose restrictions on theparameters of the Heun equation

One of these restrictions is derived by calculating thecoefficient 119860119873+1 of the term proportional to 119899119873+1 which isreadily shown to be 2 + 119873 minus 120575 Hence

120575 = 2 + 119873 (55)

The second restriction imposed on the parameters of theHeun equation is checked to be a polynomial equation of thedegree119873 + 1 for the accessory parameter 119902


With the help of the Fuchsian condition 1+120572+120573 = 120574+120575+120576we have

120574 + 120576 minus 120572 minus 1 = 120573 minus 120575 = 120573 minus 2 minus 119873 (56)


120574 + 120576 minus 1 = 120572 + 120573 minus 120575 = 120572 + 120573 minus 2 minus 119873 (58)

so that the two-term recurrence relation (51) can be rewrittenas (1198880 = 1)

119888119899 = ((120572 minus 2 minus 119873 + 119899) (120573 minus 2 minus 119873 + 119899)(120572 + 120573 minus 2 minus 119873 + 119899) 119899 119873prod119896=1

sdot 119890119896 + 119899119890119896 minus 1 + 119899) 119888119899minus1 119899 ge 1 (59)

Note that it follows from this relation since 120572 120573 and1198901 119890119873 are not zero or negative integers that 119888119899may vanishonly if 120572 is a positive integer such that 0 lt 120572 lt 2 + 119873 or 120573 isa positive integer such that 0 lt 120573 lt 2 + 119873

Resolving the recurrence (51) the coefficients of expan-sion (15)-(18) are explicitly written in terms of the gammafunctions as

119888119899 = Γ (120574 + 120576) Γ (119899 + 120574 + 120576 minus 120572) Γ (119899 + 120574 + 120576 minus 120573)119899Γ (120574 + 120576 minus 120572) Γ (120574 + 120576 minus 120573) Γ (119899 + 120574 + 120576) 119873prod119896=1

sdot 119890119896 + 119899119890119896 119899 ge 1 (60)

Here are the explicit solutions of the recurrence relation(11) for119873 = 0 and119873 = 1

119873 = 0120575 = 2 (61)

119902 = 119886120574 + (120572 minus 1) (120573 minus 1) (62)

119888119899 = Γ (120574 + 120576) Γ (119899 + 120574 + 120576 minus 120572) Γ (119899 + 120574 + 120576 minus 120573)119899Γ (120574 + 120576 minus 120572) Γ (120574 + 120576 minus 120573) Γ (119899 + 120574 + 120576) (63)

119873 = 1120575 = 3 (64)

1199022 minus 119902 (4 + 119886 minus 3120572 minus 3120573 + 2120572120573 + 3119886120574) + (120572 minus 2)sdot (120572 minus 1) (120573 minus 2) (120573 minus 1)+ 119886 (4 + 2119886 minus 4120572 minus 4120573 + 3120572120573) 120574 + 211988621205742 = 0

(65)

1198901 = minus119902 + 119886 (1 + 120574) minus 1 + (120572 minus 1) (120573 minus 1) (66)

119888119899 = Γ (120574 + 120576) Γ (119899 + 120574 + 120576 minus 120572) Γ (119899 + 120574 + 120576 minus 120573)119899Γ (120574 + 120576 minus 120572) Γ (120574 + 120576 minus 120573) Γ (119899 + 120574 + 120576)sdot 1198901 + 1198991198901 (67)

These results are readily checked by direct verification ofthe recurrence relation (11) with coefficients (16)-(18) Weconclude by noting that similar explicit solutions can bestraightforwardly derived for the expansion (21)-(25) as well

6 Discussion

Thus we have presented an expansion of the solutions ofthe Heun equation in terms of hypergeometric functionshaving the form 21198651(120572 120573 1205740 + 119899 119911) with 1205740 = 120576 + 120574 andexpansions in terms of functions 21198651(120572 120573 1205740 minus 119899 119911) with1205740 = 120574 120572 120573 For any set of parameters of the Heun equationprovided that 120574 + 120576 120574 120572 120573 are not all simultaneously integersat least one of these expansions can be applied Obviouslythe expansions are meaningless if 120572120573 = 0 since then theinvolved hypergeometric functions are mere constants andfor the solution the summation produces the trivial result119906 = 0

The applied technique is readily extended to the fourconfluent Heun equations For instance the solutions ofthe single- and double-confluent Heun equations using theKummer confluent hypergeometric functions of the forms11198651(1205720 + 119899 1205740 + 119899 1199040119911) 11198651(1205720 + 119899 1205740 1199040119911) and 11198651(1205720 1205740 +119899 1199040119911) are straightforward By slight modification equationsof more general type eg of the type discussed by Schmidt[22] can also be considered In all these cases the terminationof the series results in closed-form solutions appreciated inmany applications A representative example is the determi-nation of the exact complete return spectrum of a quantumtwo-state system excited by a laser pulse of Lorentzian shapeand of a double level-crossing frequency detuning [45] Alarge set of recent applications of the finite-sum expansionsof the biconfluent Heun equation in terms of the Hermitefunctions to the Schrodinger equation is listed in [37] andreferences therein

Regarding the closed-form solutions produced by thepresented expansions this happens in three cases 120576 = minus119873120576 + 120574 minus 120572 = minus119873 120576 + 120574 minus 120573 = minus119873 119873 = 0 1 2 3 In eachcase the generalHeun equation admits finite-sum solutions ingeneral at119873+1 choices of the accessory parameter 119902 definedby a polynomial equation of the order of 119873 + 1 The lasttwo choices for 120576 result in quasi-polynomial solutions whilein the first case when 120576 is a negative integer the solutionsinvolve119873+1 hypergeometric functions generally irreducibleto simpler functions Discussing the termination of thisseries we have shown that this is possible if a singularityof the Heun equation is an apparent one We have furthershown that the corresponding finite-sum solution of thegeneral Heun equation has a representation through a singlegeneralized hypergeometric functionThe general conclusionsuggested by this result is that in any such case the power-series expansion of theHeun function is governed by the two-term recurrence relation obeyed by the power-series for thecorresponding generalized hypergeometric function 119901119865119902

There are many examples of application of finite-sumsolutions of the Heun equation both in physics and math-ematics [46ndash56] for instance the solution of a class offree boundary problems occurring in groundwater flow inliquid mechanics and the removal of false singular points


of Fuchsian ordinary differential equations in applied math-ematics [43] Another example is the derivation of thethird independent exactly solvable hypergeometric potentialafter the Eckart and the Poschl-Teller potentials which isproportional to an energy-independent parameter and has ashape that is independent of this parameter [44] Some otherrecent examples can be found in references listed in [14]

Finally we have shown that there exist infinitely manychoices of the involved parameters for which the three-term recurrence relations governing the hypergeometricexpansions of the solutions of the general Heun equation arereduced to two-term ones The coefficients of the expansionsare then explicitly expressed in terms of the gamma functionsWe have explicitly presented two such cases

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research has been conducted within the scope of theInternational Associated Laboratory IRMAS (CNRS-Franceamp SCS-Armenia) The work has been supported by theRussian-Armenian (Slavonic) University at the expense of theMinistry of Education and Science of the Russian Federationthe Armenian State Committee of Science (SCS Grant no18RF-139) the Armenian National Science and EducationFund (ANSEF Grant no PS-4986) and the project ldquoLead-ing Russian Research Universitiesrdquo (Grant no FTI 24 2016of the Tomsk Polytechnic University) T A Ishkhanyanacknowledges the support from SPIE through a 2017 Opticsand Photonics Education Scholarship and thanks the FrenchEmbassy in Armenia for a doctoral grant as well as theAgence Universitaire de la Francophonie andArmenian StateCommittee of Science for a Scientific Mobility grant

References

[1] K Heun ldquoZur Theorie der Riemannrsquoschen Functionen zweiterOrdnung mit vier Verzweigungspunktenrdquo MathematischeAnnalen vol 33 no 2 pp 161ndash179 1888

[2] A Ronveaux EdHeunrsquos Differential Equations OxfordUniver-sity Press Oxford UK 1995

[3] S Y Slavyanov andW Lay Special functions OxfordMathemat-ical Monographs Oxford University Press Oxford UK 2000

[4] E Renzi and P Sammarco ldquoThe hydrodynamics of landslidetsunamis Current analytical models and future research direc-tionsrdquo Landslides vol 13 no 6 pp 1369ndash1377 2016

[5] M Renardy ldquoOn the eigenfunctions for Hookean and FENEdumbbell modelsrdquo Journal of Rheology vol 57 no 5 pp 1311ndash1324 2013

[6] M M Afonso and D Vincenzi ldquoNonlinear elastic polymers inrandom flowrdquo Journal of Fluid Mechanics vol 540 pp 99ndash1082005

[7] I C Fonseca and K Bakke ldquoQuantum Effects on an Atom witha Magnetic Quadrupole Moment in a Region with a Time-Dependent Magnetic Fieldrdquo Few-Body Systems vol 58 no 12017

[8] Q Xie H Zhong M T Batchelor and C Lee ldquoThe quantumRabi model solution and dynamicsrdquo Journal of Physics AMathematical and General vol 50 no 11 113001 40 pages 2017

[9] C A Downing and M E Portnoi ldquoMassless Dirac fermions intwo dimensions Confinement in nonuniform magnetic fieldsrdquoPhysical Review B CondensedMatter andMaterials Physics vol94 no 16 2016

[10] P Fiziev and D Staicova ldquoApplication of the confluent Heunfunctions for finding the quasinormal modes of nonrotatingblack holesrdquo Physical ReviewD Particles Fields Gravitation andCosmology vol 84 no 12 2011

[11] D Batic D Mills-Howell and M Nowakowski ldquoPotentials ofthe Heun class the triconfluent caserdquo Journal of MathematicalPhysics vol 56 no 5 052106 17 pages 2015

[12] H S Vieira and V B Bezerra ldquoConfluent Heun functions andthe physics of black holes resonant frequencies HAWkingradiation and scattering of scalar wavesrdquo Annals of Physics vol373 pp 28ndash42 2016

[13] R L Hall and N Saad ldquoExact and approximate solutions ofSchrodingerrsquos equation with hyperbolic double-well potentialsrdquoThe European Physical Journal Plus vol 131 no 8 2016

[14] M Hortacsu Proc 13th Regional Conference on MathematicalPhysics World Scientific Singapore 2013

[15] R Schafke and D Schmidt ldquoThe connection problem for gen-eral linear ordinary differential equations at two regular singu-lar points with applications in the theory of special functionsrdquoSIAM Journal on Mathematical Analysis vol 11 no 5 pp 848ndash862 1980

[16] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol 3 McGraw-Hill NewYork NY USA 1955

[17] L J Slater Generalized Hypergeometric Functions CambridgeUniversity Press Cambridge UK 1966

[18] W N Bailey Generalized Hypergeometric Series Stechert-Hafner Service Agency 1964

[19] N Svartholm ldquoDie Losung der Fuchsrsquoschen Differentialgle-ichung zweiter Ordnung durch Hypergeometrische PolynomerdquoMathematische Annalen vol 116 no 1 pp 413ndash421 1939

[20] A Erdelyi ldquoThe Fuchsian equation of second order with foursingularitiesrdquo Duke Mathematical Journal vol 9 pp 48ndash581942

[21] A Erdelyi ldquoCertain expansions of solutions of the Heun equa-tionrdquoQuarterly Journal of Mathematics vol 15 pp 62ndash69 1944

[22] D Schmidt ldquoDie Losung der linearen Differentialgleichung 2Ordnung um zwei einfache Singularitaten durch Reihen nachhypergeometrischen Funktionenrdquo Journal fur die Reine undAngewandte Mathematik vol 309 pp 127ndash148 1979

[23] E G Kalnins and J Miller ldquoHypergeometric expansions ofHeun polynomialsrdquo SIAM Journal on Mathematical Analysisvol 22 no 5 pp 1450ndash1459 1991

[24] S Mano H Suzuki and E Takasugi ldquoAnalytic solutions ofthe Teukolsky equation and their low frequency expansionsrdquoProgress of Theoretical and Experimental Physics vol 95 no 6pp 1079ndash1096 1996

[25] B D Sleeman and V B Kuznetsov ldquoHeun Functions Expan-sions in Series of Hypergeometric Functionsrdquo httpdlmfnistgov3111


[26] T Kurth and D Schmidt ldquoOn the global representation of thesolutions of second-order linear differential equations havingan irregular singularity of rank one in by series in terms ofconfluent hypergeometric functionsrdquo SIAM Journal on Math-ematical Analysis vol 17 no 5 pp 1086ndash1103 1986

[27] S Mano and E Takasugi ldquoAnalytic solutions of the Teukolskyequation and their propertiesrdquo Progress of Theoretical andExperimental Physics vol 97 no 2 pp 213ndash232 1997

[28] L J El-Jaick and B D Figueiredo ldquoSolutions for confluentand double-confluentHeun equationsrdquo Journal ofMathematicalPhysics vol 49 no 8 083508 28 pages 2008

[29] T A Ishkhanyan and A M Ishkhanyan ldquoExpansions of thesolutions to the confluent Heun equation in terms of theKummer confluent hypergeometric functionsrdquo AIP Advancesvol 4 Article ID 087132 2014

[30] E W Leaver ldquoSolutions to a generalized spheroidal wave equa-tion Teukolskyrsquos equations in general relativity and the two-center problem in molecular quantum mechanicsrdquo Journal ofMathematical Physics vol 27 no 5 pp 1238ndash1265 1986

[31] L J El-Jaick and B D Figueiredo ldquoConfluent Heun equationsconvergence of solutions in series of Coulomb wavefunctionsrdquoJournal of Physics A Mathematical and General vol 46 no 8085203 29 pages 2013

[32] B D B Figueiredo ldquoGeneralized spheroidal wave equation andlimiting casesrdquo Journal of Mathematical Physics vol 48 ArticleID 013503 2007

[33] A Ishkhanyan ldquoIncomplete beta-function expansions of thesolutions to the confluent Heun equationrdquo Journal of Physics AMathematical and General vol 38 no 28 pp L491ndashL498 2005

[34] E S Cheb-Terrab ldquoSolutions for the general confluent andbiconfluent Heun equations and their connection with Abelequationsrdquo Journal of Physics A Mathematical and General vol37 no 42 pp 9923ndash9949 2004

[35] T A Ishkhanyan Y Pashayan-Leroy M R Gevorgyn C Leroyand A M Ishkhanyan ldquoExpansions of the solutions of thebiconfluent Heun equation in terms of incomplete Beta andGamma functionsrdquo Journal of Contemporary Physics (ArmenianAcademy of Sciences) vol 51 pp 229ndash236 2016

[36] A Hautot ldquoSur des combinaisons lineaires drsquoun nombrefini de fonctions transcendantes comme solutions drsquoequationsdifferentielles du second ordrerdquo Bulletin de la Societe Royale desSciences de Liege vol 40 pp 13ndash23 1971

[37] T A Ishkhanyan and A M Ishkhanyan ldquoSolutions of the bi-confluent Heun equation in terms of the Hermite functionsrdquoAnnals of Physics vol 383 pp 79ndash91 2017

[38] A Ishkhanyan and K-A Suominen ldquoNew solutions of Heunrsquosgeneral equationrdquo Journal of Physics A Mathematical andGeneral vol 36 no 5 pp L81ndashL85 2003

[39] C Leroy and A M Ishkhanyan ldquoExpansions of the solutionsof the confluent Heun equation in terms of the incomplete Betaand the Appell generalized hypergeometric functionsrdquo IntegralTransforms and Special Functions vol 26 no 6 pp 451ndash4592015

[40] J Letessier G Valent and JWimp ldquoSome differential equationssatisfied by hypergeometric functionsrdquo in Approximation andcomputation (West Lafayette IN 1993) vol 119 of Internat SerNumer Math pp 371ndash381 Birkhauser Boston Boston MassUSA 1994

[41] R S Maier ldquoP-symbols Heun identities and 3 F2 identitiesrdquoin Special Functions and Orthogonal Polynomials vol 471 ofContemp Math pp 139ndash159 Amer Math Soc Providence RI2008

[42] K Takemura ldquoHeunrsquos equation generalized hypergeometricfunction and exceptional Jacobi polynomialrdquo Journal of PhysicsA Mathematical and General vol 45 no 8 085211 14 pages2012

[43] A V Shanin and R V Craster ldquoRemoving false singular pointsas a method of solving ordinary differential equationsrdquo Euro-pean Journal of Applied Mathematics vol 13 no 6 pp 617ndash6392002

[44] A Ishkhanyan ldquoThe third exactly solvable hypergeometricquantum-mechanical potentialrdquo EPL (Europhysics Letters) vol115 no 2 2016

[45] A M Ishkhanyan and A E Grigoryan ldquoFifteen classes ofsolutions of the quantum two-state problem in terms of theconfluent Heun functionrdquo Journal of Physics A Mathematicaland General vol 47 no 46 465205 22 pages 2014

[46] L Carlitz ldquoSome orthogonal polynomials related to ellipticfunctionsrdquo Duke Mathematical Journal vol 27 pp 443ndash4591960

[47] K Kuiken ldquoHeunrsquos equation and the hypergeometric equationrdquoSIAM Journal on Mathematical Analysis vol 10 no 3 pp 655ndash657 1979

[48] J N Ginocchio ldquoA class of exactly solvable potentials I One-dimensional Schrodinger equationrdquo Annals of Physics vol 152no 1 pp 203ndash219 1984

[49] G Valent ldquoAn integral transform involving Heun functions anda related eigenvalue problemrdquo SIAM Journal on MathematicalAnalysis vol 17 no 3 pp 688ndash703 1986

[50] G S Joyce ldquoOn the cubic lattice Green functionsrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 445 no 1924 pp 463ndash477 1994

[51] G S Joyce and R T Delves ldquoExact product forms for thesimple cubic lattice Green function Irdquo Journal of Physics AMathematical and General vol 37 no 11 pp 3645ndash3671 2004

[52] G S Joyce and R T Delves ldquoExact product forms for thesimple cubic lattice Green function IIrdquo Journal of Physics AMathematical and General vol 37 no 20 pp 5417ndash5447 2004

[53] R S Maier ldquoOn reducing the Heun equation to the hypergeo-metric equationrdquo Journal of Differential Equations vol 213 no1 pp 171ndash203 2005

[54] M van Hoeij and R Vidunas ldquoBelyi functions for hyperbolichypergeometric-to-Heun transformationsrdquo Journal of Algebravol 441 pp 609ndash659 2015

[55] R Vidunas and G Filipuk ldquoParametric transformationsbetween the Heun and Gauss hypergeometric functionsrdquo Funk-cialaj Ekvacioj Serio Internacia vol 56 no 2 pp 271ndash321 2013

[56] R Vidunas and G Filipuk ldquoA classification of coveringsyieldingHeun-to-hypergeometric reductionsrdquoOsaka Journal ofMathematics vol 51 no 4 pp 867ndash903 2014

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differential equations including those of more general typewhose nature outside a certain region of the extendedcomplex plane containing only two regular singular pointsis not necessary to be specified exactly Expansions involvingfunctions other than powers have been applied to the generaland confluentHeun equations bymany authorsThe ordinaryhypergeometric [19ndash25] confluent hypergeometric [26ndash29]Coulomb wave functions [30 31] Bessel and Hankel func-tions [32] incomplete Beta and Gamma functions [33ndash35]Hermite functions [36 37] Goursat and Appell generalizedhypergeometric functions of two variables of the first kind[38 39] and other known special functions have been usedas expansion functions A useful property suggested by theseexpansions is the possibility of deriving finite-sum solutionsby means of termination of the series

As far as the expansions of the general Heun equation interms of the hypergeometric functions are concerned in theearly papers by Svartholm [19] Erdelyi [20 21] and Schmidt[22] the intuitive intention was to apply hypergeometricfunctions with parameters so chosen as to match the Heunequation as closely as possible For this reason they used func-tions of the form 21198651(120582 + 119899 120583 minus 119899 120574 119911) which have matchingbehavior in two singular points 119911 = 0 and 1 These functionshave the following Riemann 119875-symbol representation

119875( 0 1 infin0 0 120582 + 119899 1199111 minus 120574 1 minus 120575 120583 minus 119899 ) (1)

Here 120582 and 120583may adopt several values provided 1 + 120582 + 120583 =120574 + 120575 (see [25]) It is clear that the functions have matchingcharacteristic exponents at 119911 = 0 and 119911 = 1 and theirbehavior does not match that of the Heun function at thethird singular point of the hypergeometric equation 119911 = infin

However it has been shown that one can also usefunctions that have matching behavior at only one singularpoint [38] Exploring this idea in the present paperwe discussthe hypergeometric expansions of the solutions of the Heunequation in terms of functions of the form 21198651(120572 120573 1205740 plusmn119899 119911)which havematching behavior (ie characteristic exponents)only at the singular point 119911 = infin These functions arepresented by the Riemann 119875-symbol

119875( 0 1 infin0 0 120572 1199111 minus (1205740 plusmn 119899) 1 minus (1205750 ∓ 119899) 120573 ) (2)

where 1 + 120572 + 120573 = 1205740 + 1205750 and the parameters 1205740 1205750 arechosen so that the Fuchsian condition for the general Heunequation (3) ie 1 + 120572 + 120573 = 120574 + 120575 + 120576 is fulfilled 1205740 + 1205750 =120574+120575+120576 Note that these functions differ also from the Jacobi-polynomials used by Kalnins andMiller whose functions canbe written in terms of hypergeometric functions of the form21198651(120582 + 119899 120583 minus 119899 ] + 2119899 119911) [23]

In the present paper we discuss several expansions interms of thementioned hypergeometric functions In generalthe coefficients of the expansions obey three-term recurrencerelations similar to those known fromprevious developments

[19ndash25] However for certain choices of the involved param-eters the recurrence relations reduce to two-term ones Inthese exceptional cases the coefficients of the expansions areexplicitly calculated The result is expressed in terms of thegamma functions

Discussing the conditions for deriving finite-sum solu-tions by means of termination of the presented series weshow that the termination is possible if a singularity of theHeun equation is an apparent one Furthermore we show thatany finite-sum solution of the general Heun equation derivedin this way has a representation through a single generalizedhypergeometric function 119901119865119902 The general conclusion is thenthat in any such case the power-series expansion of theHeun function is governed by a two-term recurrence relation(obviously this is the relation obeyed by the correspondingpower-series for 119901119865119902)2 Hypergeometric Expansions

Thegeneral Heun equation written in its canonical form is [1]

11988921199061198891199112 + (120574119911 + 120575119911 minus 1 + 120576119911 minus 119886) 119889119906119889119911 + 120572120573119911 minus 119902119911 (119911 minus 1) (119911 minus 119886)119906= 0 (3)

where the parameters satisfy the Fuchsian relation 1+120572+120573 =120574+120575+120576We introduce an expansion of this equationrsquos solutionof the form

119906 = infinsum119899=0

119888119899 sdot 21198651 (120572 120573 1205740 + 119899 119911) (4)

with the involved Gauss hypergeometric functions obeyingthe equation

11988921199061198991198891199112 + (1205740 + 119899119911 + 120575 + 120576 + 120574 minus 1205740 minus 119899119911 minus 1 ) 119889119906119899119889119911+ 120572120573119911 (119911 minus 1)119906119899 = 0

(5)

Substitution of (4)-(5) into (3) gives

sum119899

119888119899 [(120574 minus 1205740 minus 119899119911 minus 120576 + 120574 minus 1205740 minus 119899119911 minus 1 + 120576119911 minus 119886) 119889119906119899119889119911+ 120572120573119886 minus 119902119911 (119911 minus 1) (119911 minus 119886)119906119899] = 0

(6)

or sum119899

119888119899 [(119886 minus 1) (120576 + 120574 minus 1205740 minus 119899) 119911minus 119886 (120574 minus 1205740 minus 119899) (119911 minus 1)] 119889119906119899119889119911 + (120572120573119886 minus 119902)sdot 119906119899 = 0

(7)






sum119899


(10)



with





119906 = infinsum119899=0

119888119899 sdot 21198651 (120572 120573 120574 + 120576 + 119899 119911) (15)


119877119899 = (1 minus 119886) 119899 (120576 + 120574 + 119899 minus 1) (16)








[[[[[[[[[

1198760 1198771 01198750 1198761 1198772 00 1198751 1198762 1198773d d d0 119875119873minus1 119876119873

]]]]]]]]]

[[[[[[[[[[

119888011988811198882119888119873

]]]]]]]]]]=[[[[[[[[[[

0000

]]]]]]]]]] (20)



119906 = infinsum119899=0

119888119899 sdot 21198651 (120572 120573 1205740 minus 119899 119911) (21)





where

1205740 = 120574 or 120572 or 120573 (25)











(28)


sdot (120572 minus 119886 (1 + 120574) (120575 minus 1)) = 0(29)










119906119906 (0) = 31198652 (120572 120573 119890 + 1 120574 119890 119911) (33)


119890 = 119886120572120573119902 minus 119886120572120573 (34)



(35)





(36)



with


(38)


119906119906 (0) = 41198653 (120572 120573 119890 + 1 119903 + 1 120574 119890 119903 119911) (39)


119902 = 119886120572120573(1 + 119890) (1 + 119903)119890119903 (40)






= 119873+21198651+119873 (120572 120573 1198901 + 1 119890119873 + 1 120574 1198901 119890119873 119911) (42)









and 119902 = 119886120572120573 + 119886 (1 minus 120575) 120576 (43)


119906 = infinsum119896=0

(1205762)119896 ((120574 + 120576 minus 120572) 2)119896 ((120574 + 120576 minus 120573) 2)119896119896 ((120574 + 120576) 2)119896 ((1 + 120574 + 120576) 2)119896sdot 21198651 (120572 120573 120574 + 120576 + 2119896 119911) (44)


119906 (0) = 31198652 (120574 + 120576 minus 1205722 120574 + 120576 minus 1205732 1205762 120574 + 1205762 1 + 120574 + 1205762 1) (45)








1198871 119887119873 119887119873+1 = 1198901 119890119873 120574 + 120576 (50)




sdot 119890119896 + 119899119890119896 minus 1 + 119899 (51)




119896=1

sdot 119890119896 minus 2 + 119899119890119896 minus 1 + 119899 = 0

(52)



(119890119896 + 119899) + 119876119899minus1 119873prod119896=1


119896=1

(119890119896 minus 2 + 119899) = 0(53)


119873+1sum119898=0

119860119898 (119886 119902 120572 120573 120574 120575 120576 1198901 119890119873) 119899119898 = 0 (54)



120575 = 2 + 119873 (55)













sdot 119890119896 + 119899119890119896 119899 ge 1 (60)


119873 = 0120575 = 2 (61)

119902 = 119886120574 + (120572 minus 1) (120573 minus 1) (62)


119873 = 1120575 = 3 (64)


(65)




6 Discussion








Data Availability




Acknowledgments


References





























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery










sum119899


(10)



with





119906 = infinsum119899=0

119888119899 sdot 21198651 (120572 120573 120574 + 120576 + 119899 119911) (15)


119877119899 = (1 minus 119886) 119899 (120576 + 120574 + 119899 minus 1) (16)








[[[[[[[[[

1198760 1198771 01198750 1198761 1198772 00 1198751 1198762 1198773d d d0 119875119873minus1 119876119873

]]]]]]]]]

[[[[[[[[[[

119888011988811198882119888119873

]]]]]]]]]]=[[[[[[[[[[

0000

]]]]]]]]]] (20)



119906 = infinsum119899=0

119888119899 sdot 21198651 (120572 120573 1205740 minus 119899 119911) (21)





where

1205740 = 120574 or 120572 or 120573 (25)











(28)


sdot (120572 minus 119886 (1 + 120574) (120575 minus 1)) = 0(29)










119906119906 (0) = 31198652 (120572 120573 119890 + 1 120574 119890 119911) (33)


119890 = 119886120572120573119902 minus 119886120572120573 (34)



(35)





(36)



with


(38)


119906119906 (0) = 41198653 (120572 120573 119890 + 1 119903 + 1 120574 119890 119903 119911) (39)


119902 = 119886120572120573(1 + 119890) (1 + 119903)119890119903 (40)






= 119873+21198651+119873 (120572 120573 1198901 + 1 119890119873 + 1 120574 1198901 119890119873 119911) (42)









and 119902 = 119886120572120573 + 119886 (1 minus 120575) 120576 (43)


119906 = infinsum119896=0

(1205762)119896 ((120574 + 120576 minus 120572) 2)119896 ((120574 + 120576 minus 120573) 2)119896119896 ((120574 + 120576) 2)119896 ((1 + 120574 + 120576) 2)119896sdot 21198651 (120572 120573 120574 + 120576 + 2119896 119911) (44)


119906 (0) = 31198652 (120574 + 120576 minus 1205722 120574 + 120576 minus 1205732 1205762 120574 + 1205762 1 + 120574 + 1205762 1) (45)








1198871 119887119873 119887119873+1 = 1198901 119890119873 120574 + 120576 (50)




sdot 119890119896 + 119899119890119896 minus 1 + 119899 (51)




119896=1

sdot 119890119896 minus 2 + 119899119890119896 minus 1 + 119899 = 0

(52)



(119890119896 + 119899) + 119876119899minus1 119873prod119896=1


119896=1

(119890119896 minus 2 + 119899) = 0(53)


119873+1sum119898=0

119860119898 (119886 119902 120572 120573 120574 120575 120576 1198901 119890119873) 119899119898 = 0 (54)



120575 = 2 + 119873 (55)













sdot 119890119896 + 119899119890119896 119899 ge 1 (60)


119873 = 0120575 = 2 (61)

119902 = 119886120574 + (120572 minus 1) (120573 minus 1) (62)


119873 = 1120575 = 3 (64)


(65)




6 Discussion








Data Availability




Acknowledgments


References





























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery











(28)


sdot (120572 minus 119886 (1 + 120574) (120575 minus 1)) = 0(29)










119906119906 (0) = 31198652 (120572 120573 119890 + 1 120574 119890 119911) (33)


119890 = 119886120572120573119902 minus 119886120572120573 (34)



(35)





(36)



with


(38)


119906119906 (0) = 41198653 (120572 120573 119890 + 1 119903 + 1 120574 119890 119903 119911) (39)


119902 = 119886120572120573(1 + 119890) (1 + 119903)119890119903 (40)






= 119873+21198651+119873 (120572 120573 1198901 + 1 119890119873 + 1 120574 1198901 119890119873 119911) (42)









and 119902 = 119886120572120573 + 119886 (1 minus 120575) 120576 (43)


119906 = infinsum119896=0

(1205762)119896 ((120574 + 120576 minus 120572) 2)119896 ((120574 + 120576 minus 120573) 2)119896119896 ((120574 + 120576) 2)119896 ((1 + 120574 + 120576) 2)119896sdot 21198651 (120572 120573 120574 + 120576 + 2119896 119911) (44)


119906 (0) = 31198652 (120574 + 120576 minus 1205722 120574 + 120576 minus 1205732 1205762 120574 + 1205762 1 + 120574 + 1205762 1) (45)








1198871 119887119873 119887119873+1 = 1198901 119890119873 120574 + 120576 (50)




sdot 119890119896 + 119899119890119896 minus 1 + 119899 (51)




119896=1

sdot 119890119896 minus 2 + 119899119890119896 minus 1 + 119899 = 0

(52)



(119890119896 + 119899) + 119876119899minus1 119873prod119896=1


119896=1

(119890119896 minus 2 + 119899) = 0(53)


119873+1sum119898=0

119860119898 (119886 119902 120572 120573 120574 120575 120576 1198901 119890119873) 119899119898 = 0 (54)



120575 = 2 + 119873 (55)













sdot 119890119896 + 119899119890119896 119899 ge 1 (60)


119873 = 0120575 = 2 (61)

119902 = 119886120574 + (120572 minus 1) (120573 minus 1) (62)


119873 = 1120575 = 3 (64)


(65)




6 Discussion








Data Availability




Acknowledgments


References





























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery








(36)



with


(38)


119906119906 (0) = 41198653 (120572 120573 119890 + 1 119903 + 1 120574 119890 119903 119911) (39)


119902 = 119886120572120573(1 + 119890) (1 + 119903)119890119903 (40)






= 119873+21198651+119873 (120572 120573 1198901 + 1 119890119873 + 1 120574 1198901 119890119873 119911) (42)









and 119902 = 119886120572120573 + 119886 (1 minus 120575) 120576 (43)


119906 = infinsum119896=0

(1205762)119896 ((120574 + 120576 minus 120572) 2)119896 ((120574 + 120576 minus 120573) 2)119896119896 ((120574 + 120576) 2)119896 ((1 + 120574 + 120576) 2)119896sdot 21198651 (120572 120573 120574 + 120576 + 2119896 119911) (44)


119906 (0) = 31198652 (120574 + 120576 minus 1205722 120574 + 120576 minus 1205732 1205762 120574 + 1205762 1 + 120574 + 1205762 1) (45)








1198871 119887119873 119887119873+1 = 1198901 119890119873 120574 + 120576 (50)




sdot 119890119896 + 119899119890119896 minus 1 + 119899 (51)




119896=1

sdot 119890119896 minus 2 + 119899119890119896 minus 1 + 119899 = 0

(52)



(119890119896 + 119899) + 119876119899minus1 119873prod119896=1


119896=1

(119890119896 minus 2 + 119899) = 0(53)


119873+1sum119898=0

119860119898 (119886 119902 120572 120573 120574 120575 120576 1198901 119890119873) 119899119898 = 0 (54)



120575 = 2 + 119873 (55)













sdot 119890119896 + 119899119890119896 119899 ge 1 (60)


119873 = 0120575 = 2 (61)

119902 = 119886120574 + (120572 minus 1) (120573 minus 1) (62)


119873 = 1120575 = 3 (64)


(65)




6 Discussion








Data Availability




Acknowledgments


References





























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







and 119902 = 119886120572120573 + 119886 (1 minus 120575) 120576 (43)


119906 = infinsum119896=0

(1205762)119896 ((120574 + 120576 minus 120572) 2)119896 ((120574 + 120576 minus 120573) 2)119896119896 ((120574 + 120576) 2)119896 ((1 + 120574 + 120576) 2)119896sdot 21198651 (120572 120573 120574 + 120576 + 2119896 119911) (44)


119906 (0) = 31198652 (120574 + 120576 minus 1205722 120574 + 120576 minus 1205732 1205762 120574 + 1205762 1 + 120574 + 1205762 1) (45)








1198871 119887119873 119887119873+1 = 1198901 119890119873 120574 + 120576 (50)




sdot 119890119896 + 119899119890119896 minus 1 + 119899 (51)




119896=1

sdot 119890119896 minus 2 + 119899119890119896 minus 1 + 119899 = 0

(52)



(119890119896 + 119899) + 119876119899minus1 119873prod119896=1


119896=1

(119890119896 minus 2 + 119899) = 0(53)


119873+1sum119898=0

119860119898 (119886 119902 120572 120573 120574 120575 120576 1198901 119890119873) 119899119898 = 0 (54)



120575 = 2 + 119873 (55)













sdot 119890119896 + 119899119890119896 119899 ge 1 (60)


119873 = 0120575 = 2 (61)

119902 = 119886120574 + (120572 minus 1) (120573 minus 1) (62)


119873 = 1120575 = 3 (64)


(65)




6 Discussion








Data Availability




Acknowledgments


References





























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery
















sdot 119890119896 + 119899119890119896 119899 ge 1 (60)


119873 = 0120575 = 2 (61)

119902 = 119886120574 + (120572 minus 1) (120573 minus 1) (62)


119873 = 1120575 = 3 (64)


(65)




6 Discussion








Data Availability




Acknowledgments


References





























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery








Data Availability




Acknowledgments


References





























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery








































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery








Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery





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