Hypergeometric Functions Over Finite Fields

Hypergeometric Functions Over Finite FieldsAuthor(s): John GreeneSource: Transactions of the American Mathematical Society, Vol. 301, No. 1 (May, 1987), pp.77-101Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/2000329 .

Accessed: 08/12/2014 09:42

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access toTransactions of the American Mathematical Society.

http://www.jstor.org

This content downloaded from 128.235.251.160 on Mon, 8 Dec 2014 09:42:56 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=ams

http://www.jstor.org/stable/2000329?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 301, Number 1, May 1987

HYPERGEOMETRIC FUNCTIONS OVER FINITE FIELDS

JOHN GREENE

ABSTRACT. In this paper the analogy between the character sum expansion of a complex-valued function over GF(q) and the power series expansion of an analytic function is exploited in order to develop an analogue for hypergeometric series over finite fields. It is shown that such functions satisfy many summation and transformation formulas analogous to their classical counterparts.

1. Introduction and notation. Let GF(q) denote the finite field with q elements where q is a power of a prime. If X is a multiplicative character of GF(q)*, extend X to all of GF(q) by setting X(O) = 0. It is easy to see that any function f: GF(q) -> C has a unique representation

(1.1) f(x) = fat(x) + E fx (x), x

where the sum ranges over all multiplicative characters of GF(q) and 8 is the function

(1.2) 8(x)( ifx=0,

In fact, the constants f6 and fx are given by

(1.3) f = f1(0), fx = 1 f (x)x(x) q x E=GF(q)

The expression (1.1) represents a finite field analogue for the power series expansion of a function.

As an example, let Tr be the trace map from GF(q) to GF(p), set D = e2"I'P and define the Gauss sum by

(1.4) G(X) = E t E GF(q)

The Gauss sum is the analogue of the gamma function

(1.5) F(x) = txe- dt

Received by the editors January 16, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 1IT21, llLxx; Secondary 33A30,

lIG20. This paper contains parts of the author's Ph.D. thesis written under the supervision of Professor D.

Stanton at the University of Minnesota.

(C1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page

77



78 JOHN GREENE

The power series expansion for eX can be written in terms of the gamma function oo k

(1.6) ex= E x

k=O F(k + l)~ Comparing (1.4) and (1.5) one might guess that an analogous formula holds for

- Tr(x) and, in fact, it follows easily from (1.3) that

(1.7) -Tr(x) = I + q E x(x)

This analogy, which can essentially be expressed by xk - x(x), dates back at least to Jacobi (see for example [13]). Recently there has been renewed interest in this analogy. For example, in [11, Theorem 2] Helversen-Pasotto gives a result

(1.8) 1 E G(AX)G(Bx)G(CX)G(DX) q -1x

G(AB)G(AD)G(BC)G(CD) + q(q - 1)AC(-1)3(ABCD), G(ABCD)

where the term S(ABCD) is 1 if ABCD is the trivial character and 0 otherwise. This result is analogous to Barnes' lemma

(1.9) 21J 1i F(a + z)(b - z)F(c + z)F(d - z) dz

F(a + b)F(a + d)F(b + c)r(c + d) F(a + b + c + d)

where the path of integration is curved so as to separate the increasing and decreasing sequence of poles.

More examples of this analogy can be found in the references [5-11, 14 and 15]. Evans [7] has recently exploited this analogy to produce character sum analogues for Hermite polynomials and has shown that many of the classical properties carry over to the finite field case. In this paper we conduct a similar study of a character sum analogue for hypergeometric series. It is hoped that a study of special function analogues will serve as a useful tool when applied to problems surrounding character sums in the literature.

The organization of this paper is as follows. In ?2 we derive an analogue for the binomial theorem. Binomial coefficients are then defined and their elementary properties are investigated. In ?3, hypergeometric and generalized hypergeometric series are introduced. Transformations and summation theorems are given in ?4 and we give examples in ?5. In the remainder of this section we set notation.

Throughout this paper, capital letters A, B, C,... and Greek letters X, ,, ... will denote multiplicative characters. We will let 8 denote both the function on GF(q) defined in (1.2) and the function on characters

(1.10) a(A) = J 1 if A is the trivial character, 0 otherwise.



HYPERGEOMETRIC FUNCTIONS OVER FINITE FIELDS 79

The trivial character will be denoted E and the quadratic character (p. Define X by XX = E. Write Ex to denote the sum over all x in GF(q) and Ax to denote the sum over all characters of GF(q). Let J(A, B) denote the Jacobi sum

(1.11) J(A, B) = ?A(x)B(1 - x). x

Elementary properties of Gauss and Jacobi sums which we will make use of are [12, pp. 92, 93]

(1.12) G(A)G(A-) = qA(-1) -(q - 1)8(A) = qA(-1)(1 - q 8 (A)),

(1.13) J(A, B) = J(B, A)

and

(1.14) J(A,B)= G(A)G(B) +(q - 1)A B(-1 (AB). G(AB)

We will also make frequent use of the orthogonality relations for characters [12, pp. 89, 90]

(1.15) EA(x)B(x) = (q - 1)8(AB) x

and

(1.16) E X(x) = (q - 1)8(1 - x). x

2. Binomial coefficients and the binomial theorem. The binomial theorem states

(2.1) (1 + x), a X k),

where (ak) iS the "binomial coefficient" defined by

(2.2) (a) a(a - 1) ... (a-k+1) _(a + 1) * k k! 1r(k + 1)]r(a - k + 1)

A character sum analogue for (2.1) follows easily from (1.1) with f(x) = A(1 + x). The result is

THEOREM 2.3. For any character A of GF(q) and x E GF(q),

A(1 + x) = S(x) + 1 E J(A, X) X (-x)

PROOF. By (1.3),

fx = q _ E A(1 + x)x(x).

The result follows from the change of variables y = -(1 - x). El It follows from Theorem 2.3 that the finite field analogue for the binomial

coefficient is the Jacobi sum. It will prove useful to make this analogy more explicit.



80 JOHN GREENE

DEFINITION 2.4. For characters A and B of GF(q) define (A) by

(A) B(-1)J(A )

In terms of binomial coefficients, Theorem 2.3 can be written

(2.5) A(l + x) = 8(x) + q E A (x) .

Some useful properties of binomial coefficients which follow easily from properties of Jacobi sums are

(2.6) (:B) (

(2.7) (A) =B B 1)

(2.8) ( B (A)AB(-1),

and

(2.9) ()= G(A)G(J)B(1) + q-1(AB)

G(B)G ( AB ) q - G (A )__ __ _

G (B) G(AB-) +

q S()+ q (A

Some important variants of the binomial theorem are

(2.10) A(1 - x) = 8(x) + q ( AxX)x(x)

and

(2.11) B(x)AB(l- x) = q 1 BX X(x)

The result in (2.10) follows from (2.5) and (2.7). The result in (2.11) follows from (2.10) and the change of variables X -* BX.

From (2.9) we derive the following special cases for binomial coefficients

(2.12) () (A ) ( q

(2.13) (e)A(-1) + (A)

and

(2.14) q 2




Less trivial but very important are the formulas

(2.15) ( q (-I)8(A)

+ 2AB(-1)S(BC)

and

(2.16) (AB)(B) A)(A)AA(4)=

The result in (2.15) is the analogue of the well-known classical identity for binomial coefficients

(2.17)= (2 .17) ( ~~k )(n )=(k )(n - kk

The proof of (2.15) follows from a tedious but straightforward calculation using (2.9). The result in (2.16) is similar except that one must also invoke a special case of the Hasse-Davenport relation [3, p. 153]

(2.18) G(A)G(TA) = G(A2)G(T)A(4).

3. Hypergeometric series. Recall that the hypergeometric series is defined by

(3 .1) F ( a, b Ix ) = E ( a)k( b)kxk (3.1) 2

1 c

x k-O k!(c)kX

where

(3.2) (a) k = ( )=a(a + 1) ... (a + k - 1). IF(a)

More generally,

(3.3) (~ai,,a2, ...,9ar 0

= 02 (al)k(a2 )k ..(ar)k k 3 s bl b b s) kE k!(bl)k (b ) k

The most elementary attempt to define a character sum analogue for (3.3) would thus be

G(B1)G(B2) ... G(Bs) E G(AIX)G(A2X) ... G(ArX) (x)

G(A1)G(A2)*.. G(Ar) x G(X)G(BIX). G(B x) (X )

using G(AX)/G(A) as an analogue for F(a + k)/F(a). Unfortunately, this ap- proach leads to rather poor results. An alternative that gives better results is to develop an analogue for the integral representation for the hypergeometric series [2,

p. 41

(3.4) F( a, = F(c) ) t tb(I -bj tX)-a

dt 2cX IF(b)]F(c - b) o ~ tc)l- t(1 - t)



82 JOHN GREENE

DEFINITION 3.5. For characters A, B and C of GF(q) and x ? GF(q),

Al A B Cx = e(X) B(--) EB(, , l-)(lx)

y

In this definition we have dropped the normalizing constant r(c)/J(b)r(c - b) in order to obtain simpler results. The factor E(x)BC(-l)/q is chosen so as to lead to a better expression in terms of binomial coefficients.

THEOREM 3.6. For characters A, B and C of GF(q) and x e GF(q),

2F( A, B

= q 1 X(AX)( x)()

PROOF. By (2.10),

A(i1- xy) - (xy) + q AX1 x( )x(xy)-

Since e(x)B(y)3(xy) 0 for all x, y we have

2 (l A

C BX

) BC

-l E (AX )X ( x) BX ( Y),IBC(l - y )

BC(-1) ( X J(BX, BC) X (x) q -1Ix

q 1 (AX) (BC x(x)

The result now follows since

(BX _ Bx BC) (CXV

Note that the classical hypergeometric series can also be expressed in terms of binomial coefficients

37) 2 ( aS b x)=C ( k )(c + k - 1 )

More generally,

(3.8)

a( a a,, ..., an 0 a,, - 1 k a, i+k-( aO + k k-\k

n?l nk bl,..., bn )/ kLb k -bl+k1I bn b k-1)

where

(3.9) = s s bl o te bn . 1

This leads directly to the following definition.




DEFINITION 3.10. For characters Ao, A '... 'An, B1,...,Bn of GF(q) and x E

GF(q),

n+?1n B( ... = qq1 - (A x) ( AB X) * ( B X ) x(x).

Again we drop the factor C in order to obtain simpler results later on. The important special case

(3.11) 1Fo(A Ix) = e(x)(1 -x),

follows from (2.10). Generalized hypergeometric series have the following inductive integral represen-

tation [1, p. 19] (3.12)

ao, a,,...,an ) = r(bn) a0, a1,..., an-I n4-1 nV bl,.. sbnix I(an)]F(bn -an) nFn-I bl,..., bnl

*tan (1-t) bn-an dt t(1 - t)

The analogue for this result also holds.

THEOREM 3.13. For characters Ao, All ..,1 An, B1,..., Bn of GF(q) and x E

GF(q),

( AOI AI ... An AnBn(-1) AO Al, ... . An-1 n41

Bl, .. ,B n . . q LnFn-It B B xy

BAn(Y)(nBn,(A1-., PROOF. We use

(AnX) ( nX ) AnBI+1 )J(A x,4B )

-AnB/(-1) E AnX(y)AnBn(1 Y) q

in Definition 3.10 to obtain

F ( AO,All ...A = AAnB,(-1) E q (Aox) (An-lX)

*X(xy) An(Y)-nBn(1- Y),

and the result follows. O

COROLLARY 3.14. For characters A, B, C, D and E of GF(q) and x e GF(q),

(i) F2( A,B,C x

= ()BCDE(-1) E( )BCDE(-) C(y)CE(I - y)B(z)BD(1 - z)J(l - xyz), q2 y,z



84 JOHN GREENE

3i) F2( Alv B, C |X)

= E(X)BD21) ? AE(y)CE(I - y)B(z)BD(I - z)A(y - xz). q _y,z

PROOF. The result in (i) follows from Theorem 3.13 and Definition 3.5. The result in (ii) follows from (i) by the change of variables y -* l/y. El

It is clear from (3.3) that for the classical hypergeometric series, if a numerator parameter equals a denominator parameter, then the series can be written as a hypergeometric series with r and s each reduced by one. We expect a similar result in the finite field case. In addition, the series will also reduce to one of lower order if one of the numerator parameters is the trivial character. To simplify notation we adopt the following conventions: for characters A1, . . . An, B1,.:. B let

(B)

denote the product

J(Ak~

k=1 Bk!

and let

F( CA x) B

denote the series

Ft BI Bn |

Since the order of the binomial coefficients in (3.10) is irrelevant, there are five cases to consider.

THEOREM 3.15. Using the above notation,

(i) A,DI ) = i-(X)F( ACIBCX) +e(x)( A)(B)

(ii) ( ~A B IC ) A, C )AB)(B (ii) FqF -x+Bx BID D q i5 \ DB~!

(iii)

Ale,B,IC = D AD, BD,ICD F _+ A(I)(x AF x)- D,E,F A) ED, FD

q D(-I)E(x) E ) + 2 D(-\)E(x)F( B C x ) (A15)




(iv) F( A9 , ) D (A)F( BE Dx)--A(_X) C

+ q2 A(-l)E(x)F( F F--X) (B)

(V) F( A,B,C,D X) = (CE)F( A,,DX _ B (lB((B(

+ 2 BE(-1)F(A'F|)(D)

PROOF. Cases (i) and (ii) follow easily from

AX q FE

Cases (iii), (iv) and (v) follow from (2.15). For example, in (iv) the binomial coefficients

( X) (AX]) appear in the character sum for F. By (2.15)

(A)( )(B)+ q-1 A ( 1 ) C

qA F

and the result follows. 1d An important special case of Theorem 3.15 is where n = 1.

COROLLARY 3.16. For characters A, B, C of GF(q) and x E GF(q),

(i) A2F1( ~' |x) = (c)e(x) - '(x)iBC(1 -X

(ii) 2F1( 'C |x) = (A j)A(-1)C(X)A7C(1- X)

1X 1 q-1

--C(-1)E(X) + q A(-1A )(1.-x>8(C),

qX q

(iii) 2F1( 'Ai a x) = (fr)e(x) F(1i- X)

--1 A (-X) + q -1 A(-1X) (1-X) 8 B), q q

(iv) 2F1( A,lx) = -e(x)J(1- X) +s)(lBX

DEFINITION 3.17. A hypergeometric series to which Theorem 3.15 applies is called degenerate.

Another important topic to be discussed is that of symmetry. The classical hypergeometric function is invariant under any permutation of either the numerator or the denominator parameters. As Theorem 3.15 shows, that is not the case for our



86 JOHN GREENE

analogue. There are some obvious symmetries present; for example, numerator and denominator parameters are paired off in binomial coefficients in Definition 3.10 so the series is invariant under permutations of pairs of parameters:

(3.18) 3F2( A B, C x F ( A, C, B x

Similarly, special cases can occur; for example,

(3.19) 3F2 A,B ' ) =3F2( ADB,B x

We now derive the relation in general between series if parameters are permuted. We restrict our attention to transpositions. By the symmetry we know to exist, we need only consider two cases.

THEOREM 3.20. In the notation preceding Theorem 3.15,

() (AD )F( AD, E|)- q-1 D(l) (X)

Bb AlB A B,, q (i) + q2 q(-1/ Ex)8(A) AD ~ ~ A DI q E|) q (E-A x)DA

BEi (B )F E,F,Glx 2(B)(GgBF(-1)B(x>S(BE)

? q 2 BF(-1)AF AI F A Ix ),8 ( BDD

q

(Bfl( A,F,BGD - q1 ( B BE() )

? q2 BE( 1)F( A F'-D x) (CE 1.

PROOF. We use the reduction formulas of Theorem 3.15 as follows. For (i), consider the series

q ~ ~~~~~~~~~~~~~~~ E, D,

This series can be reduced by either Theorem 3.15(iv) or by (v). Setting the two expressions equal results in (i). For (ii), apply Theorem 3.15(v) in two different ways on the series




COROLLARY 3.21. If the series is nondegenerate, then

is invariant under any permutation of numerator or denominator parameters.

PROOF. Without loss of generality we may restrict our attention to transpositions of numerator parameters. Again as in Theorem 3.20 there are two cases; we will consider the second. By Theorem 3.20(ii), since the series is nondegenerate,

( _Ff AB,C, Dx =CE F A,C, B, D A

BF \ E, F, G / BE E, F,G

Thus,

( ( E ) ( F F)- ) G - E'Fx )

f BE E CE F( A, C,B, D (( BF)(E) ~F)( G) BE)F E' F'D|x

and so we must show that

C( C)B)( C)(CE) C)(B)

or

( BF) ( E) ( ) (cE E ( F Since none of the characters involved is trivial, by (2.9) we have

(CF B jr(CA G(B)G(C) (CE ( B jr C kBF!kE IkV F= G(E)G(F)G(BE)C(BF) G(CB) BE Elk EJ

Case (i) is similar. [ The factor ( ) in Corollary 3.21 arises because we did not normalize hypergeomet-

ric series in Definition 3.10. The reason for dropping this factor is that the results in Theorem 3.15 and 3.20 become even worse if this constant is incorporated. It also leads to simpler results in the next section. The complications in Theorems 3.15 and 3.20 might lead one to question the definition we are using for hypergeometric series. The results would be much simpler if one were to use a definition involving Gauss sums instead of Jacobi sums. The justification for our definition comes in the next section when we give transformations and evaluation theorems for hypergeometric functions. These theorems would become intractible using a definition in terms of Gauss sums because the binomial theorem with Gauss sums instead of Jacobi sums is more complicated than Theorem 2.3.

4. Transformations and summation theorems. Classical hypergeometric series satisfy a large number of transformation identities. Kummer gave a list of 24 solutions to the differential equation for the hypergeometric series

(4.1) x(1 - x)y" +(c-(a + b + I)x)y' - aby = 0.



88 JOHN GREENE

Any three such solutions form a linearly dependent set and the resulting linear dependence relations give transformations for hypergeometric series. For an account of these transformations and a list of Kummer's 24 solutions see [18, pp. 283-286 or 4, pp. 105-108].

Kummer's techniques for obtaining solutions for (4.1) involved changes of variables which do not alter the form of the differential equation. We do not have an analogue for (4.1) so we must resort to other means in obtaining transformations. Our techniques are based on the binomial theorem and changes of variables in Definitions 3.5 and 3.10 or in Theorem 3.6.

THEOREM 4.2. For characters A, B, D, C, E of GF(q) and x E GF(q),

F(A,B, I \ x -Dx BD, AD,ICD (i) FI=Dx)I_

D,E J D, ED I

(ii) F AB Ex) = ABDCE(-l)A(x)F A, AD AE_+ 1

PROOF. We change variables in Definition 3.10,

F( A,B C ) -q 1 AX)(BX) Cx(x)

For (i), replace X by DX, for (ii), replace X by AX and appeal to property (2.8) for binomial coefficients. O

We remark that the change of variables in (i) can be carried out for any denominator parameter, and the change of variables in (ii) can be done for any numerator parameter to obtain similar transformations. The result in (ii) is the analogue for the classical result on reversing the summation of a terminating hypergeometric series [17, p. 48]

(3)n+1 n ( bl b|x

_ (a,)m *,, (an)m (\m F( -m,1-b-m .1-bn- m|\ (bl)m (bn) mXJmn+l?lf 1-a1-,m,...,1-an-m x

It is interesting to note that because the character sum has only a finite number of terms, the analogy is strongest with terminating hypergeometric series. In fact, if there is a difference between terminating and nonterminating cases, our analogue will be with the terminating case of a formula. This phenomenon can be seen in the first part of the following theorem.

THEOREM 4.4. For characters A, B, C of GF(q) and x E GF(q),

(i) 2F1( A, x = A(-1) 2F1( AB B1 -x

+A( A- I _ x)_ X) (x),




(iii) 2F1(A ]x)=B(1Ax)2F,( ci A,ii ) +A( 1 (B) 8(1_ ),

(iv) 2F1( A' A

B) = C(-1)CAB (1 -x)2F1( C, c x )

+A(-l)( B)8(1- x)

PROOF. We change variables in the definition

2 AF , ' X = (x) q E B(y)BC(l - y)X(1 - xy).

For (i), we set y =z/(z - 1). Then we have 1 - y = 1/(1 - z) and 1 - xy =

(1 - (1 - x)z)/(1 - z). Thus,

(A, B

B

= ?(X) (-) q B(z)B(z - 1)BC(l - z)A(1 - z)A(1 -(1 - x)z)

=E(x)A(-A) ( B(z)AC(I - z)A(1 -(-x)z). q

The result now follows from the observation that ?(x) = ?(1 - x) + 8(1 - x) -

8(x). For (ii), (iii) and (iv) make the changes of variables

z 1-z y = 1- z, Y 1 -x?xz and Y =1 respectively. E

Theorem 4.4, parts (ii) and (iii) are analogues for Pfaff's transformation [17, p. 31]

) F( a b x) (1X) F1 c x -I1

part (iv) is an analogue for Euler's transformation [17, p. 101

(4.6) 2F1( ab a x )=(1X)c-a-b2F, c-ac-b x)

Part (i) is an analogue for the transformation [17, p. 341 (4.7)

a, b b- -)=F(1 ?a+b-c)F(1-c) 21ak b 2F1( 1 + a + b

_ cI-x) = r(I + a- c)F(l+b - c) 2F a, c

?(I + a + b - c)F(c-1) 1-C F( 1+ a-cJl + b-cx) + -F(a)r(b)

2 12-C c



90 JOHN GREENE

To strengthen the analogy, note that if a is a negative integer, say a= -n, then 1/r(a) = 0 so (4.7) becomes, in the terminating version,

(4.8) 2F1( -n, b = (cLb) n2F1( -n b - x 1

which agrees well with 4.4(i). It may be pointed out that if one includes the trivial y - z then there are 24

changes of variables of the type used in the proof of Theorem 4.4. This gives rise to 23 transformations (24 if one includes the identity transformation). These 24 changes of variables and the resulting transformations correspond to Kummer's 24 solutions. We have listed the most important transformations. The others may be obtained by various compositions of the transformations given in Theorem 4.2, in the case of the 2F1, with the transformations in Theorem 4.4.

THEOREM 4.9. For characters A, B and C of GF(q),

2F1( AS B I1 = A (-1) A-C)

PROOF. Set x = 1 in Theorem 4.4. Alternatively, one could set x= 1 in the definition of a2Fj:

2FI ( Il) q )B(y) BC(1 -y) A(I -y) y

=- q Ji(B ABC) = (-IA ) q

9 A \ CJ

Theorem 4.9 is the analogue of Gauss's evaluation [2, p. 2]

(4.10) 2F1( a,b i)= f(c)F(c-ab)

In fact, if F is nondegenerate, then putting in the proper normalization yields

(B )2 (A 9 B I1 A (-1) ( _B ) ( B)

-A(-1) G(C)G(BC) G(C)G(CAB) G(CA)G(ABC) G(CA)G(CB)

Definition 3.5 leads to other summation theorems as well. For example, if we set x = -1 we have

2F1( A' -I) = BC(-) Z B(y)BC(1 -y)A(1 + y). c q v~~~




If C = AB this becomes

2F1( A- I) - (1 B(y)A(1 - y2) ( AB I ) q

y

A(-I) LB(y) q E A X(y2)

=A (-I) AX -BX2(y).

The inner sum is zero unless BX2 is trivial so we are lead to

A B 0 if B is not a square,

(4.11) (AB| ) i\(A) +(KA if B = C2.

This result is the analogue of Kummer's theorem [2, p. 9]

(4.12) F ( a, b 1) F(I + b - a)F(I + Ilb) (4.12) 2F (a,?ba i)= 2(+-)(+b I + b _ a F(I + b)F(I + lb - a

The terminating form of (4.12) is

(0 if n is odd, (4.13) 2

a, - ~- ,1 (N) N (3I( 1- a-n (a N)

if n = 2N.

Using Theorem 4.4 we are lead to

A if B is not a square,

(4.14) 2FI( A2) 2 A(-I){(c) +(PC) if B = C2,

A if B is not a square,

(4.15) 2FI A 2 ) =A(-2)(C) +( ) if B = C2.

A large number of quadratic transformations exist for classical hypergeometric series. A partial list of quadratic transformations can be found in [4, pp. 64-67], which also gives necessary and sufficient conditions for the existence of a quadratic transformation in terms of the parameters of the function. A similar result holds for hypergeometric series over GF(q), but it is not our intention to examine the problem in such depth. Instead we will derive a single quadratic transformation by way of example.

THEOREM 4.16. For characters A and B of GF(q) and x E GF(q), ( 0 if B is not a square,

A-Bx) =AS(1 +Ax)k(c) () if B =C2



92 JOHN GREENE

PROOF. Using the definition for a 2F1 we have

A~B \ A (-1) 2FI ABX = ?(X) q ) B(y)A(l - y)J(1 - xy)

A (-1) = -(X)X( 1 B(y)A(l -y(l + x-xy)).

q y

By the binomial theorem,

-y(l + x - xy))

= (Y(l + x - xy)) + q A

> (AX)X(Y)X(l + x - xy).

Using this in (4.16) and changing y to y/x gives

(4.17) 2F1( AB x B X) +(-1 AB =

Ax)( q -

* (AX) BX (x)BX(y)X(I + x -y).

X,y X

Since 1 = -(I + x) + S(1 + x), (4.17) becomes (4.18)

2F1(A-B ) = A B( X) + 8(1 + x) 1 x ( X)B(_1)BX2(Y)

+?(I + X ) -1 L (x) BX (x)BX(y)X(1 + x -y). q -1 Y \x

The second term on the right-hand side was evaluated in obtaining (4.11). To evaluate the third term make the change of variables y - (1 + x)y to obtain

AB\ A (-1)/1 + 0 if B is not a square,

AB ) q x )+ (+ )yA )?+ (CA) ifB= C2,

( X ) A-1) 1X

E BX(y)X(1 - y).

The inner sum on this last term is

J(BX, X) = qX (-I)(X)-

Using this and making the change X - BX gives the desired result. [ Theorem 4.16 gives the best analogue available for the classical transformation [4,

p. 113]

(4.19) a, b+ x) = (1-x) 2F1( 1t;2 b 4x)




It is somewhat disappointing, however, that the major term on the right-hand side of (4.16) does not, strictly speaking, represent a hypergeometric series. The problem lies in the term (Bx). If B is a square then we can exploit (2.16) to improve the analogy.

THEOREM 4.20. For characters A and B of GF(q) and x E GF(q),

2F1( A-B2|X) = 8(1 + x)[(A) ?( A)]

+A(-1) q - e(x)e(1 + x)8(pB)

+ q1 ( pj) A(-1)B 1 _ X)(x))?(1 + x)8(AB).

(A)(B) A(1)B (1+X2F( B ( X)2

PROOF. It follows from (2.16) that

(4.21) (B X = (BX)(B2X)( T) BX(4).

Consequently, if B is replaced by B2 in Theorem 4.16 we obtain

2( AB: ) = 8(1 + X)[(A) + ( B)] + A(-1) B2(1 X)

+B 2(1 + x)B(4)(j) q 12X 2 (I + )2

The character sum on the right-hand side is a degenerate 3F2 which when reduced to a 2F1 gives rise to the desired result. [1

Given the quadratic transformation in Theorem 4.20, others may now be obtained by using the 2F1-linear transformations. More information on quadratic transformations over finite fields can be found in [9, pp. 40-55].

Transformations for higher order hypergeometric series also exist. In fact, 3F2(1) transformations follow from 2F1(x) transformations. For example, if both sides of the result in Theorem 4.4(i) are multiplied by (DE(-1)/q)D(x)DE(1 - x), then summing on x gives

(4.22) DE -1) 2F1 A 'Jx) D(x)DE(1 -x)

q x c

AE (1E 2F1( | 1 - x)D(x)DE(I-x). q -

A)C

If we replace x by 1 - x on the right-hand side, then by Theorem 3.13 we have

(4.23) 3F2k ABjD 1) =AE(-1)3F2( A, r AB F5 1



94 JOHN GREENE

Similarly, from Theorem 4.4, parts (ii), (iii) and (iv) we obtain

(4.24) 3F2 A

C Ell) =AC(-1)3F2( C

' D

) C9E ~~C, B DE.

(4.25) 3F2 A, BD = B(-1)3F2( CA, B, D

and

(4.26) 3F2 A C, E ll) =AB(-1)3F2( Ci C ABCCE

respectively. There exists a 3F2 quadratic transformation [17, p. 90],

(4.27) a,2( 'l + c- a,l + c - b

=(1-X-c3F(1C 2c 2+?c,I1+ c- a -b -4x~ 1+ c - a, 1 + c - b (1-x)2

As might be expected, a similar result holds for the finite field case.

THEOREM 4.28. For characters A, B and C of GF(q) and x E GF(q),

F(A, B, CIX) = q-AC(-1),(x)S(b)

+ 2 BC(-I),-(x),(l + x)S(A)- 2 A (-x)S (AB C) q q

+I1 ABC(-1)( ) + x() + I(A) BC(-1)C q

O if C is not a square

+8(1~~~ - xAB(I D (BD ) (pD ( mBD 2

- X)A Q4){()(BD A TAD if C=D2

+BC(-1)C(1-X) q 1 (C I )(jXk ABC _ x)21

SKETCH OF PROOF. We proceed similarly to the proof of Theorem 4.16, beginning with the analogue of the integral representation, Corollary 3.14(i),

(4.29) 3F2 ( x

= (X) 2 ) E C(y)B(1 -y)B(z) AB C(I - z)(1 - xyz). q y,z




Expanding A(1 - z)A(1 - xyz) = A(1 - z(1 + xy - xyz)) by the binomial theorem and taking z to 1 - z gives

A CA- B = e(x) A - q C-X) E C(y)

*B(- y)BC(z)BX(l - z)X(1 + xyz)

= AC(-1) q_1 X (Ax)()F BC C Cx C q - I x x CX, BC

again by Corollary 3.14(i). When the 3F2 on the right-hand side is reduced, the main term is

ABC(-1) q 1 Y( X)( X(-1)2FI( x' x)

One may now apply the 2F1-quadratic transformation producing the main term

AB( q-1 A(AX)( BX)()( - q q - I x x x ~q -1

z(C ) ( Cj)( 2)

=A()C(1 - X) c~ F A B, s I) AB_) q )-1E I )+E _ X)2 3 2 e' CA|1

The above 3F2 reduces to a 2F1(1) which can be evaluated by Theorem 4.9 to obtain the summation on the right-hand side of Theorem 4.28. O

As was the case with Theorem 4.16, more can be said if the appropriate character is a square. For simplicity we will only state the result in the nondegenerate case.

COROLLARY 4.30. If A, B, cpC, CA and C2AB are all nontrivial, then

3F2 2 A B- c2- Ix)

= - (1-X)AB((-[) [(A ) A B C + B 1)6(1 + D

+ (A)(_ AB(-c)c ( 2 p C, C2AB -4x A 99C I -x1\ C2A, c2B (1 _X)2j

PROOF. Similar to that of Theorem 4.20. O We close this section with some important 3F2(1) evaluations. In the classical case

there are four major summation theorems. These are Saalschutz's theorem [17, p. 49]



96 JOHN GREENE

Dixon's theorem [17, p. 51]

(4.32) 3F2( 1 + a- b,I + a - c I

-F(I + -a)F(I + a - b)F(I + a - c)F(I + 'a - b - c)

F(I + a)F(I + la - b)F(1 + 'a - c)F(I + a - b - c) Watson's theorem [17, p. 54]

(4.33) 3F2( -, 2(1 + a + b),2c

- F()F(l + c)F(1(1 + a + b))F(I(I - a - b) + c) F(-(I + a))F(I(I + b))F(1(1 - a) + c)F(I(I - b) + c)'

and Whipple's theorem [16, p. 54]

(4.34) F( 1 b , 1 - b + 2c

7f(b)r(l - b + 2c) 22c-lF('(a + b))Fr((I + a - b + 2c))Fr((1 - a + b))Fr((2 - a - b + 2c))

We give analogues for each of these.

THEOREM 4.35. For characters A, B, C and D of GF(q),

( D, ABC D ) C-)DA qc)-D(l(B PROOF. Recall Theorem 4.4(iv),

(4.36) 2F1( A'B X) = C(-1)CAB(1- X)2F1( CA, C X

+A(-1)(I B ) s(1- X). We calculate the coefficient of D(x) on both sides of (4.36). The coefficient of D(x) in

2F1( A, B ) is

q (AD (BD q-1\ D\CD}

Since

8(1 - x) = qx ),

the coefficient of D(x) in A(-1)( -P)8(1 - x) is I A (- 1)(A-)

q -

By the binomial theorem, the coefficient of D(x) in X(x)CAB(I - x) is

q CAB DX(-I)




Collecting terms, we have

q__ AD B _ (I q 2(ci7x(C3x (CAB x q-1 D)(CD) q - I x CX) D)-J )

+ q A

or

(AD (CD A(-1)1 - =ABC(-1)q X CT3 )-

~DjCDj q AC q-1I \xJ\ CX CABD X

The result now follows by relabeling the parameters. O


CA, CB q

-q -

AC(-1)8(B)q - 2 A(-1)8(CAB) q2 q2

0 if C is not a square,

+AB(-I) (D BD ) (D A )( BD 2

A -I +

A - Iif C= D.

ABD A \TABDI

PROOF. Set x = 1 in Theorem 4.28. 0 Theorems 4.35 and 4.37 are the analogues of Saalschitz's theorem and Dixon's

theorem, respectively. The theorems of Watson and Whipple are obtained from Dixon's theorem by using the 3F2(1) transformations.


(i) 3F2( AC, B2 |)

= q-12 C(-1)8(A) 2 AC(-1)6(B) -q 2 C(-1)8(ABC) q q q

O if C is not a square,

q ~-1+1q -II q fCD2 AlA)(ABD )(A- AABD

(ii)

3 A2 A2C- = 8q2 (A)-q 2 6(B) 2 q 2A(-1)S(CA7

O if BC is not a square,

+B I )(AB-D\ + AD ( BD 2 A -1+1 A

- Iif BC= D. ABD A TABD



98 JOHN GREENE

PROOF. For (i), apply transformation (4.25) to the hypergeometric series in Theorem 4.37 and replace A by A. For (ii), apply transformation (4.23) and replace C by BC. O

5. Remarks. Several of the results in this paper have previously appeared in the literature under different guises. For example, the result [11, Theorem 2; 15, Theorem A2],

(5.1) 1 E G(AX)G(B-)G(C-)G(D-) x

G(AB)G(AD)G(BC)G(CD) + q(q - l)AC(-l)8(ABCD), G (ABCD)

is equivalent to Theorem 4.9 via (2.9). Also, results [5, (4)] and [5, (5)] are essentially two different formulations of Dixon's theorem, Theorem 4.37.

It is the author's hope that techniques involving hypergeometric series over finite fields will provide a useful and elementary set of tools for dealing with problems related to character sums. As an example, see [10] in which the character sum

E T(xY(G + x)(1 + y)(x + y)) x,y

is evaluated over GF(P). This evaluation can be used to obtain an explicit count of

the number of points on the surface (X2 + 1)(y2 + 1)(X2 + y2) = Z2 over GF(P) and confirms a conjecture of Evans [8, p. 370].

Even the binomial theorem, Theorem 2.3, can be quite useful. For example, the

Jacobsthal sum, Hn(a), in our notation is defined by

(5.2) Hn(a) = E p(x)Tp(a + Xn). x

A classical result involving the Jacobsthal sum is [16, p. 231]

d-1

(5.3) Hn(a) = 2(a)j(-1) j( J2j+ 1J

j=O

where d = gcd(n, q - 1) and 4 is a character of order 2d. This result is easily proved by elementary techniques but it follows almost immediately from orthogonality and the binomial theorem. We have

E T(x>)(a + Xn) = p(a)E T(x)P(1 + aj) x x

= 99 (a) Eiq,- (a )nX

q - 1 x (,)x~)p~lx x,x

by Theorem 2.3. By orthogonality, the x-sum is 0 unless x is an odd power of 4, which establishes the result.




The theory presented in this paper is by no means complete. Three notable transformations for which analogues have not been given are the balanced 4F3(1) transformation [2, p. 56]

(5.4) 4F( a

bf C,

(f -C)n(g - n4Fe -a,e- b, c, -n

(f)n(g)n 4 e, 1+c-f-n, 1+c-g-nJ'

provided e + f + g = 1 - n + a + b + c, the transformation of a nearly poised 4F3(1) to a 1-balanced special 5F4(1) [2, p. 33]

(5.5)

43( 2+a-b, l+a-c, wI)

(w-a),, I(+a-w, -m, a - 1 +a-b-c|

(w)e -(l+a-w-m),-(2+a-w-m),l+a-b, l+a-c ,

and Whipple's transformation of a very well poised 7F6(1) to a 1-balanced 4F3(1) [2, p. 25]

(56) 7 6( a 1+ a-b, 1+a-c, 1+a-d, 1+a-e, 1+a-f|)

r(i + a - d)F(1 + a - e)F(1 + a -f)r(1 + a - d - e-f) r(l + a)r(I + a - e -f)1(1 -- a - d-f)P(1 + a - d- e)

4 +a-+-c, d, e, + I +-b- I d+a-b, e+a-c, d+e+f-a)- '

provided the left-hand side converges and the right-hand side terminates. Two evaluation theorems for which analogues have not been given are the evaluation of a very well poised 5F4(1) [17, p. 56]

(5-7) s4a, I + 2, b, c, -m (I (+ a),(l +a -b - c),n l 21a, I + a-b, I +a- c, I +a+m| (I +a -b),,(l +a - c)m

'

and Dougall's theorem [17, p. 56]

(5.8) (E6 a, 1 + 2 b, c, d, e, -m 7 6

2a, I+ a-b, l+a-c, l+a-d, l+a-e, l+a+ml

(1 + a)m(l + a - b - C)m(l + a - b - d)m(1 + a - c -d)

(1 + a - b)m(l + a - c)m(1 + a - d)m(1 + a - b - c - d)m'

provided 1 + 2a = b + c + d + e - m. Analogues for the results in (5.4) and (5.5) are easily accessible. To obtain an

analogue for (5.5) multiply both sides of the quadratic transformation, Theorem 4.28, by (MW(-l)/q)M(x)MW(I - x) and sum on x. If the series involved are



100 JOHN GREENE

nondegenerate, the resulting expression is

(5.9)

A, B C =

ABCW( -1) M ()+BCW(-1) (A)MWC) ( CA, CB, Wl q (AB) W )

B( 1) q (CX2 x AB CX) Mx2JX\I

The series on the right-hand side of (5.9) is the best analogue available for

( cpx/U, V CAB, M, CW

5Jt AC, BC, VCMW, TCMW )

if neither C nor CMW is a square. To obtain an analogue for (5.4), multiply two transformations of type 4.4(iv) to obtain (5.10)

2F1 A, B D, D Ex=AD(-1)( B- )F8(1_x)

+ CF(-l)FCABDE (1-X)2F1( Cii CB X)2FI( D

FFx)

If we pick the parameters so that FCABDE(1 - x) = -(1 - x)= 1 - 6(1 - x),

then equating coefficients of G(x) gives

(5. 11) DEF(-1)q

- 1 4F3( C -DI-G

q-1 4 (C C, GDF, GEF

+AD(-I) B j( Ej AD(-I) ( Cb)(FE q-1 VAC}DF} q- Alk D}

or after relabeling the parameters, (5.12)

F3 A, B, C, DI = ADG(-1) B B CA D ADG(-1) (E3(bDG' 43 E, F, Gl q AE D FX q k AI CF}

B, C, ~i)ADG(~1) ( BC EBAJ\CPD

+CDE(-1)4F3( FA, , C, DG)

provided ABCD = EFG. These techniques do not seem to be good enough to give analogues for (5.6)-(5.8) and, in fact, there is no known analogue for these results.

REFERENCES

1. R. Askey, Orthogonalpolynomials and specialfunctions, SIAM, Philadelphia, Pa., 1975. 2. W. Bailey, Generalized hypergeometric series, Cambridge Univ. Press, Cambridge, 1935. 3. H. Davenp6rt and H. Hasse, Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen

Fallen, J. Reine Angew. Math. 172 (1934), 151-182. 4. A. Erdelyi, Higher transcendentalfunctions, Vol. 1, McGraw-Hill, New York, 1935.




5. R. Evans, Identities for products of Gauss sums over finite fields, Enseignement Math. 27 (1981), 197-209.

6. , Character sum analogues of constant term identities for root systems, Israel J. Math. 46 (1983), 189-196.

7. , Hermite character sums, Pacific J. Math. 122 (1986), 357-390. 8. R. Evans. J. Pulham and J. Sheehan, On the number of complete subgraphs contained in certain

graphs, J. Combin. Theory Ser. B 30 (1981), 364-371. 9. J. Greene, Character sum analogues for hypergeometric and generalized hypergeometric functions over

finite fields, Ph.D. thesis, Univ. of Minnesota, Minneapolis, 1984. 10. J. Greene aand D. Stanton, A character sum evaluation and Gaussian hypergeometric series, J.

Number Theory 23 (1986), 136-148. 11. A. Helversen-Pasotto, L'identite de Barnes pour les corps finis, C.R. Acad. Sci. Paris Ser. A 286

(1978), 297-300, 12. K. Ireland and M. Rosen, A Classical introduction to modern number theory, Springer-Verlag, New

York, 1982. 13. C. Jacobi, Uber die reisteilung und ihre Anwendung auf die Zahlentheorie, J. Reine Angew. Math. 30

(1846), 166-182. 14. N. Koblitz, The number of points on certain families of hypersurfaces over finite fields, Compositio

Math. 48 (1983), 3-23. 15. W. Li and J. Soto-Andrade, Barnes' identities and representations of GL(2). Part I: Finite field case,

J. Reine Angew Math. 344 (1983), 171-179. 16. R. Lidl and H. Niederreiter, Finite fields, Encyclopedia of Mathematics and Its Applications (G.-C.

Rota, ed.), Vol. 20, Addison-Wesley, Cambridge, Mass., 1983. 17. L. Slater, Generalized hypergeometricfunctions, Cambridge Univ. Press, Cambridge, 1966. 18. E. Whittaker and G. Watson, Modern analysis, Cambridge Univ. Press, Cambridge, 1947.

DEPARTMENT OF MATHEMATICS, SOUTHERN ILLINOIS UNIVERSITY, CARBONDALE, ILLINOIS 62901



Documents

Hypergeometric Functions Over Finite Fields