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Generalized exponential andlogarithmic polynomials withstatistical applicationsRamalingam Shanmugam aa University of Colorado at Denver, 1100 FourteenthStreet , Denver, Colorado, 80202, U.S.A.Published online: 09 Jul 2006.

To cite this article: Ramalingam Shanmugam (1988) Generalized exponential andlogarithmic polynomials with statistical applications, International Journal of MathematicalEducation in Science and Technology, 19:5, 659-669, DOI: 10.1080/0020739880190502

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INT. J. MATH. EDUC. SCI. TECHNOL., 1988, VOL. 19, NO. 5, 659-669

Generalized exponential and logarithmic polynomials withstatistical applications

by RAMALINGAM SHANMUGAM

University of Colorado at Denver,1100 Fourteenth Street, Denver, Colorado 80202, U.S.A.

(Received 4 March 1986)

In many statistical discussions, especially in data analysis, the idea ofpolynomials plays a key role. For example, Dwyer [1] employed polynomials toexpress factorial moments of discrete distribution in terms of cumulative totals.Traditionally, polynomials are derived using the difference operator method (see[2], p. 134]). In this article, using the differential equation approach as analternative method, we obtain generalized exponential and logarithmic poly-nomials, and find their special cases appearing in statistical signal-noise models.

1. IntroductionWe can categorize polynomials into two types: exponential and logarithmic.

Exponential polynomials E(k,hlt h2, •. -,hk) are introduced by Bell (see [2], p. 134);they are defined as

]-\} (1.1)

where

H(u)= £ *,«' (1.2)iSl

Analogous to the above definition, the logarithmic polynomial L{k, h1,h2 hk) isdefined as

' ( 1 - 3 )

Expanding the right side of (1.1) as a Taylor's series, we find

{exp[tf(M)]-l} =,-H=vW

i21k»i k\

where P(k,n,hlth2,.. .,hk) is called a potential polynomial (see [2], p. 134), and iszero when k<n. Hence, we may rewrite

aa k uk

,huh2,...,h)— (1.4)

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660 R. Shanmugam

A comparison of (1.4) with (1.1) reveals that an exponential polynomial is a linearcombination of potential polynomials, and vice versa.

Proceeding in a similar manner, we obtain

ln{l+H(u)}=fj £(-iy-i ( l--l)!

xP(k,i,huh2 hk)^- (1.5)

and a. comparison of (1.5) with (1.3) establishes a linear relationship betweenlogarithmic and potential polynomials.

In this sense of linear relationships, we discount a potential polynomial as adifferent category.

The above polynomials are derived (see [2]) using the difference operatormethod. Needless to say, there are difficulties in the difference operator methodwhen the polynomials are to be extended. To avoid such difficulties, we resort to, inthis article, a differential equation approach as an alternative method. We definegeneralized exponential and logarithmic polynomials in section 2, and investigatetheir properties. In section 3, we find special cases of our generalized polynomialswhich have applications in statistical signal-noise models.

2. Definitions and properties of generalized exponential and logarithmicpolynomials

For a real-valued a and non-negative integers k and n such that k^n, we define ageneralized exponential polynomial as

DxE(k,n,cc,gltg2 gk;hlth2,...,hk)

xE(k-i,n,cc,gug2,.. .,gk-r,hji2,.. .,hk-i) (2.1)

and a generalized logarithmic polynomial as

DxL{k,n,z,gug2,.. .,gk\huh2, ...,hk)

xL(k-i,n,cc,gl,g2,...,gk_i;h1,h2,...,hk-i) (2.2)

where Da indicates the derivative with respect to a; mu, m2i,... are non-negativeintegers such that

F(i,j) is Stirling numbers of the first kind (see [2] for the definition and properties ofStirling numbers); gug2,.. .,gk are coefficients of «' in expansion

G(«)=f f t«' (2.3)iS=l

and hl,h2,.. .,hk are as defined in (1.2).

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Generalized exponential and logarithmic polynomials 661

Note that both polynomials E(-) and L(-) are zero when k < n. Many properties ofour polynomials are identifiable from their exponential generating functions (e.g.f.).Let 4>E{-) and (j>L{-) be their e.g.f.s. That is, with a proper convergence interval for u,

4>E(.') ~ 4>E(n, <x,u,gltg2,...; h1,h2,...)

= fjE(k,n,*,gu...,gk,hu--;h)uklk\ (2.4)JcSsn

and

4>L(-) = 4>L(n,«,«.gugi»• • •; hu h2> • • •)

00

= £ L(k,n,a,gug2>.. .,gk,huh2 hk)ukjk\ (2.5)

Jfcs=n

To obtain a closed form for (/>(•), we proceed as follows. Due to definition (2.1) wenote that

Dx(j)E(n, <x,u,gug2,...,huh2,...)

(5= 1 1 = 5 1 l]-

x X E(m,n><*,gi>g2 ^;^i,A2,...,A»!

=-z ii>\

x <pE(n,x,u,gug2,...; hu h2,...)«'/«! (2-6)

To solve the differential difference equation in (2.6), we employ the known z-transformation technique. Let

^Ei-) = ^E^>u,z,gug2,.. .;huh2t...)

00

= E <t>E(n,a.,u,gug2,:. .,huh2,.. .)zn (2.7)

Then it is easy to see, using (2.6), that

x il/E(cc,u,z,gltg2,...>huh2,.. .)u'li\ (2.8)

A solution of our differential equation in (2.8) is

l}] (2.9)

for some function/(•)• An identity off(z,u,h1,h2,...) could be easily traced downfrom an initial value of E{k, n, a.,gug2,.. .,gk; hlt h2,..., hk) for w = 0. In the absenceof knowing such an initial value, we proceed, alternatively, as follows. Equating (2.7)

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662 R. Shanmugam

and (2.9), we note that

00

Y <l>E(n,ct,u,gl,g2,.. .;huh2,...)

(2.10)

Now, expressing f(z,u,huh2,...) in a Taylor's series, and substituting it in (2.10),we obtain

CO

£ <pE(n,cc,u,gug2,.. .,huh2,...)

= £ (exp[-ot{exp[G(«)]-l}-l])0x[irj{z ,u,huh2 , . . .)]|«-o

which is satisfied by

f(z, u, hlt h2,.. .) = exp[*(exp{exp[H(u)]-1} -1) ] (2.12)

Substituting (2.12) into (2.10) and evaluating at # = 0, the nth derivative with respectto z of the identity, we obtain

<l>E(-) = <f>E(f',<x,u,gug2,.. .;huh2,...)

= exp[-a{exp[G(«)]-l}]

x(exp{exp[ff(«)]-l}-l)"/n! (2.13)

Proceeding in a similar manner, we get a closed form for </>£,(•); it is

4>L(n, a, u,gug2,...; hu h2,...)

! (2.14)

Comparisons of (2.13) and (2.14) with (1.1) and (1.3), respectively, reveal thatwhen <x = 0, our polynomials E(k,n,a.,gug2,.. .,gk;huh2,.. .,hk) andL(k,n,u,gl,g2,.. .,gk; hu h2, .. .,hk) are nothing but Bell's exponential and logarith-mic polynomials, respectively. While Bell's polynomials are always integer valued,our polynomials are either integer or fractional valued depending upon a.

Property 1 If there exist functions

and

such that

H(M) = ln{l+^(M)} (2.15)

and

]} (2.16)

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Generalized exponential and logarithmic polynomials 663

then expression (2.13) becomesCO

£ E(k,n,a.,gug2 gk;hlth2;.. .,hk)

(i, n)[l +B(u)r[A(u)]ilil (2.17)

i & n * ̂ i

xP(k,i,a,bub2,.. .,bk;aua2,.. .,ak)uklk\ (2.18)

= £ £5(i,fi)

xP(k,i,cc,bub2 bk;aua2,...,ak)uklk\ (2.19)

where S(j,n) is a Stirling number of the second kind and P(k,i,a.,bltb2,.. .,bk;aiy a2,..., ak) is our generalized potential polynomial. See [2] for details of Stirlingnumbers.

Hence, a linear relationship between generalized exponential and potentialpolynomial is

E(k,n,u.,gug2,.. .,gk;huh2,.. .,hk)

x P(k, i, cc,bub2,..., bk; aua2,..., ak) (2.20)

Property 2 If there exist functions

and

such that //(«) = (exp[C(«)] — 1} and G(M) = {exp[Z)(w)] — 1}, then expression (2.14)becomes

00

£ L(k,n,a,g1,g2,...,gk;hi,h2,...,hk)uklk\

00 00

~ Z £ -F(''w)p(*'''<*-,d1,d2,...,dk\cuc2,...,ck)uklk\

= f fjF(i,n)P(k,i,a,d1,d2,...,dk;cuc2,...,cak)uklk\

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664 R. Shanmugam

where F(i,n) is the Stirling number of the first kind, and P(-) is-a generalizedpotential polynomial as denned in (2.17) and (2.18). Hence, a linear relation betweenour generalized logarithmic and potential polynomial is

L(k,n,a,gltg2,..,,gk;huh2 hk)

k

= X F(i,ri)P(k,i,a,d1,d2,...,dk;cuc2,.. .,ck) (2.21)

Property 3 Both generalized exponential and logarithmic polynomials possessthe additive convolution property with respect to n and a. That is,

n2)E(k + +,gug2 gk\huh2,...,hk)

= "Z [[)E(l,ni,<*ugug2,--;gk;h1,h2,...,hk)

xE(k-l,n2,a.2,gltg2,...,gk;huh2,...,hk) (2.22)

and

L(k,n1+n2,0L1 + cc2,gi,82>--->8k'>hi>h2>'-->hk)

)L(l,nua1,gug2,.. .,gk;huh2 hk)

xL{k-l,n2,<x2,g1,g2,...,gk;huh2,...,hk) (2.23)

In the next section, we investigate special cases of our polynomials, and theirapplications.

3. Statistical applicationsCase 1 Consider

() .g> = : = ht

for i = l , 2 Then from (1.2) and (2.3), note that

G(0) = l n ( l + 0 ) = //(0) (3.1)

Substituting (3.1) in (2.13), we obtain

! = expafl(expO-

which implies that if Y and (Z1, Z2,..., Zn) are independent Poisson and n identicalzero-truncated (positive) Poisson random variables with parameters cx.9 and 0,respectively, then the probability distribution of the mixture

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Generalized exponential and logarithmic polynomials 665

X= Y+Z1 +Z2 +... + Zn involves the generalized exponential polynomial. That is,

x,n, -a ; l , - | , i , . . . , x

07*! (3.2)

, . . . ; M = 1 , 2 , . . . ; a^O; and 9>0.There are many situations in real life where discrete data are collected only in the

form of signals plus noise. For an example, when data on radioactivity are collectedby a Geiger counter, the signal is observed only in the presence of noise, and it leadsto the necessity of a study on probabilistic properties of a convolution model like theone in (3.2). How data get contaminated in a communication system is illustrated byWoodward [3].

Case 2 Now, choosing differently for g as

gt= t (-1)*"Vtf-WF(i,jW. (3-3)

and keeping ht as in (3.1), we note

= I (-iy-V(/-D! Z *X«,i)07«!

(3.4)

Substituting (3.1) and (3.4) in (2.13), we obtain

E(x,n, -*,r,r-{\ +r),.. .,gx; 1, -

And hence,

x£(*,«, -a,r, ~ d +0, • • ..*»; 1, - i i • • - Z " ^ ' ^g*/*! (3-5)

= w , « + l , . . . ; w = l , 2 , . . . ; oc^O; r >0; 6>0;-where X= Y+Z1+Z2+ ...Zn; Yand Zt, i = l,2 n are independent binomial (roc, 0) and positive Poisson (0)random variables, respectively.

Case 3 Consider

i (3.6)

for i= l , 2 Note that |.F(i,./)| is the absolute value of Stirling numbers of the first

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666 R. Shanmugam

kind. The sign of F(i,j) is (-l)i+J. Then,

6»)s} (3.7)

Substituting (3.7) and (3.1) in (2.13), and proceeding as before, we obtain

t f — IV" 1

• Fl x n a. - t - - ( * • x - -1 1 i. r,\ x, /I, a., i, i

where 3C = M,M + 1,...; w = l ,2 , . . . ; <x>0, s>0, O<0<1; X=Y+Zl+Z2 + ...Y and Zj, i = l , 2 « are independent negative binomial (as,6) and positivePoisson (6) random variables, respectively.

Case 4 Choosingg( as in (3.1) but hi as

we obtain H(0) = ln{l —5ln(l —6)} and G(0) = ln(l +6), which upon substituting in(2.13) yield

iX£f*,«,-o1;l1-ii...,|I;Jl--J W (3.9)

for x = n, n + \,...; n = l ,2 , . . . ; a^O, 5>0, O<0<1, whereX= Y+Zt +Z2 +.. . + Zn; YandZltZ2,...,Zn are independent Poisson (a, 6) and nidentical positive negative binomial (s, 6) random variables.

Now, selecting only gf differently as in (3.3) and (3.6), we obtain the followingcases, respectively.

Case 5 Convolution of binomial and positive negative binomial randomvariables.

(3.10)

for x = n,n + l,...; a^O, r>0 , M = 1 , 2 , . . . ; O<0<1; X=Y=Z1+Z2 + .where Yand Zit i = 1, 2, . . . , n, are independent binomial (roc, 6) and positive negativebinomial (s, 9) random variables.

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Generalized exponential and logarithmic polynomials 667

Case 6 Convolution of negative binomial and positive negative binomialrandom variables.

xEL,n,a,-s,-s(\+S-,...,gx-s,s\\-^\,...,h\xjx\ (3.11)

for x = n,n+\,...; n = l , 2 , . . . ; a^O, s>0, O<0<1; where X=Y+Z1+Z2

+.. . + Zn, Y is a negative binomial random (sec, 6) variable, Zl(i=l, 2 n) arepositive negative binomial (s, 6) random variables, and are all independent.

Choosing

so that (see (3.4))

(3.12)

j differently, as in (3.1) or (3.3) or (3.7), we obtain the following results.Case 7 Convolution of Poisson and positive binomial random variables.

E\x,n, - a ; 1, - | , i .. .,gx;r, -T- (3.13)

* = n,n + l , . . . ; « = l ,2, . . . ; a > 0 ; r>0 ,Case 8 Convolution of binomial and positive binomial random variables.

x Elx,n, -a ; r, -r(\ +^\ .. .,gx; r, -T-(\ +r),.. .,h\xjx\ (3.14)

, n= 1,2Case 9 Convolution of negative binomial and positive binomial random

variables.

E(X, n,a;s, - , M + I j , . . .,gx- r, -Y-{\ +r),..., hx)8xlx\) (3.15)

for * = M,M + 1,...; a>0; s>0; 0>O; w = l,2Case 10 Convolution of Poisson and logarithmic random variables. Choosing

/?i=l/i'! (3.16)

gi= t S(i,j)W (3-17)

where S(i,j) is a Stirling number of the second kind, we note

f (3.18)

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668 R. Shanmugam

and

G(6)=fjgiei

= {exp[exp[0]-l]-l} (3.19)

Substituting (3.18) and (3.19) in (2.14), we obtain, after simplifications,

( ) / « ! (3.20)

for je = M,w + l , . . . ; « = l , 2 , . . . ; a^O; O<0<1.Case 77 Convolution of binomial and logarithmic random variables. Selecting

hi as in case 10, but gt differently as

where C(i,j, r) is Charalambides' [4] C-number. Values of C(i,j, r) could be easilycomputed using an expression in Charalambides [4]. Then,

= t t C(f,y,r)0'/ili ̂ 1 j"» 1

= 1 f;C(i,/,r)0'/|-!& i i &

l]-l} (3.22)

Now, substituting (3.22) and (3.18) in (2.14), we obtain

X=x] = {\ +0)-ra[ln(l +6)]-nn\

xL[x,n, -a ; l , ^ z ! L + 2r(2r-l),.. .,fo; l.^r,^,.. .,^)B*IXI (3.23)\ r 2! 3! x!/

Case 12 Convolution of negative binomial and logarithmic random variables.

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Generalized exponential and logarithmic polynomials 669

Selecting ht as in case 10, but£j differently as

a=ts(y,»)/ii (3.24)

where S(i,j,s) is a number orthogonal to Charalambides' C-number. Values ofS(i,j,s) could be computed easily from an expression in Charalambides [4]. Then,

= Z t 5(«,y,*)0'/i!

0)-s-l]-l} (3.25)

Substituting (3.25) and (3.22) in (2.14), we obtain the probability function of theconvolution of a negative binomial and n logarithmic random variables. That is,

xL(M,ct;l,|2S2

+1J & ; 1,1,1,...,1V/*! (3.26)

where x = n, n +1,...; « = 1, 2, . . . ; a^O; s>0; and 0 < # < l .We have noticed in all the cases discussed thus far that generalized exponential

and logarithmic polynomials are built in positive discrete signal-noise probabilitymodels. Choosing the input for the polynomials differently gives rise to differentmodels. Because both polynomials are additive with respect to their arguments a. andn (see property 3 in section 2), each one of our twelve discrete signal-noise models isinfinitely divisible with respect to parameters a and n. The implications of infinitedivisibility are being studied now, and will be reported elsewhere.

References[1] DWYER, P. S., 1940, Ann. Math. Stat., 11, 66.[2] COMTET, L., 1974, Advanced Combinatorics (New York: Reidel).[3] WOODWARD, P. M., 1953, Probability and Information Theory with Applications to Radar

(Oxford: Pergamon Press).[4] CHARALAMBIDES, C, 1977, SIAM J. appl. Math., 33, 279.

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