Certain generating functions of Hermite-Bernoulli-Legendre polynomials Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 10. № 2 (2018). С. 118-126.

УДК 517.9

CERTAIN GENERATING FUNCTIONS OF HERMITE-BERNOULLI-LEGENDRE POLYNOMIALS

Abstract. The special polynomials of more than one variable provide new means of analysis for the solutions of a wide class of partial differential equations often encountered in physical problems. Most of the special function of mathematical physics and their generalization have been suggested by physical problems. It turns out very often that the solution of a given problem in physics or applied mathematics requires the evaluation of an infinite sum involving special functions. Problems of this type arise, e.g., in the computation of the higher-order moments of a distribution or while calculating transition matrix elements in quantum mechanics. Motivated by their importance and potential for applications in a variety of research fields, recently, numerous polynomials and their extensions have been introduced and studied. In this paper, we introduce a new class of generating functions for Hermite-Bernoulli-Legendre polynomials and study certain implicit summation formulas by using different analytical means and applying generating function. We also introduce bilateral series associated with a newlv-introduced generating function by appropriately specializing a number of known or new partly unilateral and partly bilateral generating functions. The results presented here, being very general, are pointed out to be specialized to yield a number of known and new identities involving relatively simpler and familiar polynomials.

Keywords: 2-variable Hermite polynomials, Generalized Bernoulli numbers and polynomials, 2-variable Legendre polynomials, 3-variable Hermite-Bernoulli-Legendre polynomials, summation formulae, generating functions.

Generalized and multivariable forms of the special functions of mathematical physics have witnessed a significant evolution during the recent years. In particular, the special polynomials of two variables provided new means of analysis for the solutions to large classes of partial differential equations often encountered in physical problems. Most of the special function of mathematical physics and their generalization have been suggested by physical problems.

To give an example, we recall that the 2-variable Hermite Kampe de Feriet polynomials Hn(x,y) [3] defined by the generating function

N.U. Khan, T. Usman, Certain Generating Functions of Hermite-Bernoulli-Legendre Polynomials.

©N.U. Khan, T. Usman 2018. Поступила 24 мая 2017 г.

N.U. KHAN, T. USMAN

Mathematics Subject Classification: 33B10, 33C45, 33C47, 33C90

1. Introduction

(1.1)

are the solution of the heat equation

Q Q2

Qy f(x> y) = Q~2fy^ 0) = x'n.

The higher order Hermite polynomials sometimes called the Kampe de Feriet polynomials of

order m or the Gould-Hopper polynomials H%™\x, y) are defined by ([11, Eq, (6,3)])

[—]

H^\x, y) = n\ £ J, x , ^ k\(n — mk)\

m

n

In

exp(xt + ytm) = Y,Hnr)(x, y)- (1.2)

£-/ ri I

n=0

are the solution of the generalized heat equation [7]

Q Qrn

~dyf(x, y) = (x, y), f(x> 0) = xn.

We also note that

Hn\x, y) = (x + y)n, Hn2)(x, y) = Hn(x, y), where Hn(x, y) denotes the 2-variable Hermite Kampe de Feriet polynomials defined by (1.1).

We recall that the 2-variable Legendre polynomials Sn(x, y) and Rn(x, y) are given by Dattoli et al. [8]

[ - ]

[ 2 ] xkyn-2k

Sn(x, y) = n% (n — 2k)\.(k!)2], (h3)

and

^ (_i)n-k xn-kqk

Rn(x-y) = (n )2 £ !m ^, M

k=0

respectively, and are related with the ordinary Legendre polynomials Pn(x) [14] as

+_

2

Pn(x) = Sn (-,x) =Rn(^, ^)

From equation (1.3) and (1.4) we have

[ - ]

x[ 2 ]

Sn(x, 0) = n\[(^, Sn(0, y) = yn, Rn(x, 0) = (—x)n, Rn(0, y) = yn.

The generating functions for two variable Legendre polynomials Sn(x, y) and Rn(x, y) are given by [8]

^ fn tT'

eytCo(—xt2) = J2Sn(x, y)n\,, Co(xt)Co(—yt) = Y.Rn(x, v)~^2,

n=0 \ n=0 ( !)

where C0(x) is the 0-th order Tricomi function [14]

Co(x) = ]T (-1)rx

( \)2

r=0 y '

We note the following link between the Hermite Kampe de Feriet polynomials Hn(x, y) and the 2-variable Legendre polynomial Sn(x, y) (see [6]):

Hn(y, —D-1) = Sn(x, y),

where Dx 1 denotes the inverse of derivative operator Dx := Jx and is defined in such a way that

1 l'x

D-n[f (x)} = (-- (x - 0n-lf

(n — 1)! Jo

so that for f(x) = 1, we have

xn

D-n{1} = n -

n!

In view of the equations (1,1) and (1.2), we note the following relation:

H(2){x, -= Hn(x, -0 = Hen(x)

where Hen(x) denotes the classical Hermite polynomials [1].

We recall that the 3-variable Legendre-Hermite polynomials sHn(x,y, z) are defined by the series [10]

[ - ]

and are specified by the following generating function:

^ jn

exp(yt + zt2)Co(-xt2) = V]sHn(x,y, z) — .

n!

n=0

The classical Bernoulli number Bn, Bernoulli polynomials Bn(x) and their generalization Bla](x) (ofreal

or complex) of order a are usually defined by means of the following generating functions ([1], [2], [4], see also [15])

(^) =i>n, (F-t) exi=^B"(x)S,

^ / n—n \ ' n—n

t \ X ^ ~ t I t \ \ ^ ^ / \ t

n

n=o n=o

and

/ + \a ^ +n

- 1 n=o n!

The Bn are rational numbers and in particular Bn(0) = Bn(0) = Bn.

The generalized Hermite-Bernoulli polynomials of two variables hB^(x, y) are introduced by Pathan [12] in the form

/ + \a ^ +n

ext+yt2 = Y,HBia)(x, y)n, (1.6)

- 1 n=o n!

which are essentially generalization of Bernoulli numbers, Bernoulli polynomials, Hermite polynomials and Hermite-Bernoulli polynomials HBn(x, y) introduced by Dattoli et al. ([9], p.386 (1.6)) in the form

n

_L_ ext~2 = £hB„(X, y) -

- n=o

The special polynomials of more than one variable provide new means of analysis for the solutions of a wide class of partial differential equations often encountered in physical problems. It turns out very often that the solution of a given problem in physics or applied mathematics requires calculating of an infinite sum involving special functions. Problem of this type arise, for example, in the computation of the higher-order moments of a distribution or while calculating the transition matrix elements in quantum mechanics. In [5], Dattoli showed that the summation formulae of special functions often encountered in applications ranging from

electromagnetic process to combinatorics can be written in terms of Hermite polynomials of more than one variable.

In this paper, we derive the generating function for the Hermite-Bernoulli-Legendre polynomials sB<if>(x,y, z) in terms of certain multi-variable special polynomials. Also we find some implicit summation formulae by using different analytical means and applying generating functions. We also introduce bilateral series associated with the newlv-introduced generating function by appropriately specializing a number of known or new partly unilateral and partly bilateral generating functions.

2. A NEW CLASS OF HERMITE-BERNOULLI-LEGENDRE POLYNOMIALS

We define the Hermite-Bernoulli-Legendre polynomials sB^(x, y, z) by means of generating function as follows:

/ + \a ™ +n

e^2C0(-xt2) = Y.sB^(x,y, z)L

^ ' n=0 '

or equivalentlv, by the series

(2.1)

sBia)(x,y,z) = y V 2k(x;y) zk ' ,

s n v ,y, j ^ ^ (m - 2k)\k\(n — m)! '

m=0k=0 v ' y '

where Sn(x, y) are the 2-variable Legendre polynomials in (1.3), As x = 0, (2.1) reduces to

/ + \a ^ 4-n

hrr^ ^ = E»Bn°'(v, =) L (2.2)

^ ' n=0

or equivalentlv (see [13], p.682)

m f \

sB^(0,y, z) = HB^(y, Z) = ^ )B{na}mHm(y, Z) ,

n=0 '

where hB(n\y, z) are the 2-variable generalized Hermite-Bernoulli polynomials in (1.6). Letting a = 0, (2.1) yields

^ fn

eyt+^Co(-xt2) = YjsHn(x,y, z)- (23)

n!

n=0

or equivalentlv

^ v2^ (-1)r-2k(x)r-2k(y)n-rzk

n\

sB{") (x, y, z) = sHn(x, y, =

(n - r)\k\(r - 2k)\

=0 k=0

where sHn(x,y, z) are the 3-variable Hermite-Legendre polynomials in (1.5).

3. Implicit summation formulae involving Hermite-Bernoulli-Legendre

polynomials

In order to find implicit summation formulae involving Hermite-Bernoulli-Legendre polynomials sBn\x,y, z), the same consideration as developed for the Hermite-Bernoulli polynomials in Pathan et al. [13] holds as well. First we prove the following results involving Hermite-Bernoulli-Legendre polynomials sBn\x,y, z).

Theorem 3.1. The following implicit summation formulae for Hermite-Bernoulli-Legendre polynomials sB<>n") (x,y, z) holds true:

k'1 / u \ / i

sBk+ (x, v, z)= jr(kn)(l) (v - y)n+p sBk+ xPxn(x, y, z) (3.1)

n,p=0 ^ ' ^ '

Proof We replace t by t + u and rewrite the generating function (2,1) as

/ + + u \a ^ +k I

e'<'+")2Co(-x(i + u)2) = ex'<' « £ sB&(x-=) J U

V / k7=0 ! !

equation, we get

£ sBt+ (x,y, Z)l-U = £ sBl+ (x,v, z)%U (3.2)

k,l=0 ' ' k,l=0 ' '

Bv expanding exponential function (3,2) we arrive at

^ [(v-y)(t + u)]N ^ B(a) tkul = ^ B(a) tk ul

-N- z^ sBk+l(X- y, ZK\T\ = sBk+l(X- V- ,

=0

which by using formula ([15, Eq, (2)

N' ^ s k+iy k' l' ^ s k+iy ' ' 'k\l\

N=0 k, =0 k, =0

t f (n ^ = ± ^+») n m

N' m ml

N=0 n,m=0

in the left hand side becomes

Y. ^â- £ sBk+)(X. V, Z)%u = £ sBiï(X, v, ku (3.3)

^_n -i 7_n 7_r\

Now replacing k bv k — n, I bv I — p and using Lemma 1 in [15] in the left hand side of (3,3), we get

oo oo

(V - y)n+P _ ( tk u = tkul

^ ^ n\p\ sB(+-n-V(X,y. Z'Jj~n)' (t-p)'. = SB(+ (X,V. Z)k'.n

n,p=0 k,i=0 1 \ j \ rj

u

the required result, □

By taking l = 0 in Eq, (3,1), we immediately deduce the following result.

Corollary 3.1. The following implicit summation formula for Hermite-Bernoulli-Legendre polynomials SB^ (x,y, z) holds true:

sB^(x,v, z) = £ C n ) (v - y)nsBk-n(x,y, z).

n=0

Replacing v by v + y and setting x = z = 0 in Theorem 3,1, we get the following result involving Hermite-Bernoulli-Legendre polynomial of one variable:

sBit (v + y) = jt(ni)(lp) (v)m+n sB$ _p_,n(y),

n,p=0 ^ ' ^ '

whereas by setting v = 0 in Theorem 3,1, we get another result involving Hermite-Bernoulli-Legendre polynomial of one and two variable

sBkt(x, z) = (n)(lp) (—y )n+msBk^l-P-n(x,y, z).

n,p=0 ^ ' ^ '

Along with the above result, we will exploit extended forms of Hermite-Bernoulli-Legendre

(a).

polynomial sBn. )(x, y) by setting z = 0 in the Theorem 3,1 to get

sBtl (x, v) = jt(kn)(lp) (v — y)n+msBki-P-n(x, y)

A straightforward expression of the HBina (y, z) is suggested by a special case of the Theorem

x = 0

hB^v , Z) = jt(kn)(lp) (v — y)n+mHBit-p-n (y, z),

n,p=0 ^ ' ^ '

where HB(jk+l(y, z) are the 2-variable generalized Hermite-Bernoulli polynomials in (1,6),

Similarly, a straightforward expression of the sHn(x,y, z) is suggested by a special case of

a = 0

sHk+i (x,v, z) = (kn)(l) ( — V)n+msHk+i—p—n(x-, y, z~),

n,p=0 ^ ' ^ '

where sHk+ (x,y, z) are the 3-variable Hermite-Legendre polynomials in (1,5),

Theorem 3.2. The following implicit summation formula for Hermite-Bernoulli-Legendre polynomials sB^ (x,y, z) holds true:

sBla)(x,y + v,z + u) = ( m ) sBia—)m(x,y, z)Hm(v,u), (3.4)

m=0 ^ '

where Hm(v,u) are the 2-variable Hermite polynomials in (1.1).

Proof. We replace y by y + v and z by z + u in (2,1), use (1,1) and rewrite the generating function as

jrsB(a)(x,y + v,z + u) n = {( 7—iyeyt+zt2C0(—xf)^ ,

n=0 00

jn '— j-n — j-rn

n n n n m

n=0 n=0 m=0

Now, replacing n by n — m and comparing the coefficients of tn, we arrive at (3,4), □

Theorem 3.3. The following implicit summation formula for Hermite-Bernoulli-Legendre polynomials sBia'1 (x,y, z) holds true:

B(a) (x vz) = y* (-1)m(-x)mHB^2m(y, Z) n\

sBn (x,y, Z) = ^ -( 2m)\(m\)2-' (3'5)

m=0

Proof. Using the generating function (2,1), we have

tn ( t

(n — 2m)\(m\)2

^ +n / + \a

n=0 n - 1

£sWx,V, 1 £ = { £hWv, *) 1} {£ i-1)^} -

n=0 ^n=0 J U=0 v ' J

Replacing n by n - 2m and comparing the eoeffieients of tn, we obtain (3,5), □

Theorem 3.4. The following implicit summation formula for Hermite-Bernoulli-Legendre polynomials sB<*^ (x,y, z) holds true:

sB<")(x,y, z) = V V (-1) (x) Bn-7((y -U)HZ(3.6)

s n ¿¡¡m (m - 2k)!(n -m)!(k!)2 y '

Proof By exploiting the generating function (1.1), we can write Eq, (2,1) as

( / i \ a •) ^ rn ™ +m ™ ( 1)k( 2)k

{ (^l t'C0{-xt2) = (V -U)n £ Hm(u, Z) m £ (-T>(k-2t >

^ ^ ' ) n=0 ' m=0 ' k=0 ^ ''

Replacing m by m - 2k, we get

œ ,; œ | [ 2 ^ u ( \( T\k{ \k

^(X.y,,) = -u)m z|E(tIk-L^

n=0 m=0 I k=0

n n - m

r(»)( ) = V" iv V" (-1)k(-x)kBin-m(y - u)Hm-2k(u. z) 1

2_^sBn (X,y. z)m (n-m)'(m - 2k)'.(k'.)2 \

n=0 n=0 I m=0 k=0 y ' y ' y ' I

We equate the coefficients of the like powers of tn and this leads us to (3.6). □

By employing the same technique used in the proof of above theorem, from (1.6) and (2.1), we can obtain semi-addition formula for sB<\x,y, z) and the known addition formula for B<\x) (see, e.g., [[16], p. 84, Eq. (25)]) which are given in the following theorem.

Theorem 3.5. The following implicit summation formula for Hermite-Bernoulli-Legendre polynomials sB<") (x,y, z) holds true:

sB<"+l)(x,u + v, z) = it(nk) sB<âk (x,u, z)Bf\v) (3.7)

k=0

and

B<na+l3\u + v) = it(l) B<:lk (u)Bf(v) (3.8)

k=0

Proof In view of the relation (3.8) is just a special case of (3.7). □

sB<na)(0,y, 0) = B<«)(y),

m

4. Generating functions of the Hermite-Bernoulli-Legendre polynomials

involving bilateral series

We consider the function

, \ a

(a) n(a) / „, „.. „. „ i\ . I ^ \ „s—wt

Vla) = Vla)(x,y,w, z; s, t) := es-wt/s+yt+zt2 Co(-xt2)

Expanding e s-wt/s and using (2,1), we get

~ „ m ™ / wf\K . ™ ,N

Vla) = Eh e -W h EsB^(x,y, Z) - (4,)

M—n JT—O V ' N—n

We replace the summation indices M and N in (4.1) by K + N = n and M — K = m, respectively and we rearrange the triple series:

tt tt n i \K

X—V (—W)K

„m+n \ \ J

v(a) = £ ^„^ K + 'K;{n — K) s^—Kz);

m=—— n=m* K=0

hereinafter m* = max{0,—m} m e Z, Thanks to the absolute convergence of the series involved, we are led to the generating function

a

t » °-w:

(¡TT!yes-wt/s+yt+zt2 Co(-xt2)

m n (- w) ( a)

m=-<x n=m* K=0

Some special cases of (4,2) are demonstrated in the following corollaries. Corollary 4.1. Setting a = 1 in (4-8), we get

' t \ — — n (_K)K

j \ p s—wt/s+yt+zt2 C (_.2) = V^ V^ mtn V^ _( K)_

e* — 1) U0( x )= ^ ^ S ^ K!(m + K)!(n — K)!

v ' vy-t —_iVN nrt —vy-t * Tf — O ^ ' ^ '

m=-rx n=m* K=0

(o)

Corollary 4.2. Setting a = 0 and using sByn)(x,y, z) = sHn(x,y, z) in (4-2), we get

oo oo

^-wt/s+yt^(—xt,)=Ti Mfr + —¡(n — K) (•»• *

m=—— n=m* K=0

where sHn—K(x,y, z) are the 3-variable Hermite-Legendre polynomials given in (1.5).

x = 0

K

' + \a <x <x n (_w)K

1 \ s-wt/s+yt+zt2 = ST^ ST^ „m.n _( w)__Bla) (

e^ \) = ^ (m + K)\(n-K)\ HBn-K (

/ m=-<x n=m* K=0

where HBniK(y, z) are the 2-variable generalized Hermite-Bernoulli polynomials in (1.6).

СПИСОК ЛИТЕРАТУРЫ

1. L.C. Andrews. Special functions for engineer and mathematician. Macmillan Co., New York (1985).

2. T.M. Apostol. On the Lerch zeta function // Pacific. J. Math. 1:2, 161-167 (1951).

3. P. Appell and J. Kampé de Fériet, Fonctions Hypergéométriqués et Hypersphériques: Polynomes d'Hermite Gauthier-Villars, Paris (1926).

4. E.T. Bell. Exponential polynomials // Ann. Math. 35:2, 258-277 (1934).

5. G. Dattoli. Summation formulae of special functions and multivariable Hermite polynomials // Nuovo Cimento Soc. Ital. Fis. В 119:5, 479-488 (2004).

6. G. Dattoli. Monomility, orthogonal and pseudo-orthogonal polynomials // Inter. Math. Forum. 13:1, 603-616 (2006).

7. G. Dattoli. Generalized polynomials, operational identities and their application // J. Comput. Appl. Math. 118:1-2, 111-123 (2000).

8. G. Dattoli, P.E. Ricci and C. Cesarano. A note on Legendre polynomials // Int. J. Nonl. Sci. Numer. Simul. 2:4, 365-370 (2001).

9. G. Dattoli, S. Lorenzutta and C. Cesarano. Finite sums and generalized forms of Bernoulli polynomials // Rend. Mat. Appl. Ser. VII. 19:3, 385-391 (1999).

10. G. Dattoli, A. Torre, S. Lorenzutta and C. Cesarano. Generalized polynomials and operational identities // Atti. Accademia. Sci. Torino Cl. Sci. Fis. Mat. Natur. 134, 231-249 (2000).

11. H.W. Gould and A.T. Hopper. Operational formulas connected with two generalization of Hermite polynomials // Duke Math. J. 29:1, 51-63 (1962).

12. M.A. Pathan. A new class of generalized Hermite-Bernoulli polynomials // Georgian Math. J. 19:3, 559-573 (2012).

13. M.A. Pathan and W.A. Khan. Some implicit summation formulas and symmetric identities for the generalized H ermite-Bernoulli polynomials // Mediterr. J. Math. 12:3, 679-695 (2015).

14. E.D. Rainville. Special functions. Chelsea Publishing Company, New York (1971).

15. H.M. Srivastava and H.L. Manocha. A Treatise on generating functions. John Wiley and sons, New York,(1984).

16. H.M. Srivastava and J. Choi. Zeta and q-Zeta Functions and Associated Series and Integrals Elsevier Science Publishers, Amsterdam (2012).

Nabiullah Khan,

Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh-202002, India. E-mail: nukhanmath@gmail.com

Talha Usman,

Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh-202002, India. E-mail: talhausman.maths@gmail. com