M. A. Pathan. UNIFIED (p, q)-bernoulli-hermite POLYNOMIALS

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1 F A S C I C U L I M A T H E M A T I C I Nr DOI:101515/fascmath M A Pathan UNIFIED (p, q)-bernoulli-hermite POLYNOMIALS Abstract The Concepts of p-bernoulli numbers B n,p and p-bernoulli polynomials B n,p (x) are generalized to (p, q)-bernoulli numbers B n,p,q and (p, q)-bernoulli polynomials B n,p,q (x), respectively Some properties, generating functions and Laplace hypergeometric integral representations of (p, q)-bernoulli numbers B n,p,q and (p, q)-bernoulli polynomials B n,p,q (x), are established Unified (p, q)-bernoulli-hermite polynomials are defined by a generating function which aid in proving the generalizations of the results of Khan et al [8], Kargin and Rahmani [7], Dattoli [4] and Pathan [9] Some explicit summation formulas and some relationships between Appell s function F 1, Gauss hypergeomtric function, Hurwitz zeta function and Euler s polynomials are also given Key words: Bernoulli polynomials and Bernoulli numbers, generating functions, Gauss hypergeomtric function, Hurwitz zeta function and Euler s polynomials AMS Mathematics Subject Classification: 11B68,11B83, 33C05, 33C65 1 Introduction The Pochhammer s symbol or Appell s symbol or shifted factorial or rising factorial or generalized factorial function [13] is defined by (b, k) (b) k Γ(b + k) Γ(b) where b is neither zero nor negative integer and the notation Γ stands for Gamma function Generalized Gaussian hypergeometric function A F B [15,p42(1)] of one variable is defined by (1) AF B [ a1, a 2,, a A ; b 1, b 2,, b B ; ] z A F B (a 1, a 2,, a A ; b 1, b 2,, b B ; z)

2 126 M A Pathan k0 (a 1 ) k (a 2 ) k (a A ) k z k (b 1 ) k (b 2 ) k (b B ) k k! where denominator parameters b 1, b 2,, b B are neither zero nor negative integers and A, B are non-negative integers and we assume that variable z takes on complex values The Appell function F 1 of two variables is defined by [14] (2) F 1 F 1 (a, b 1, b 2 ; c; z 1, z 2 ) m,n0 (a) m+n (b 1 ) m (b 2 ) n z m 1 zn 2 (c) m+n m! (3) F 1 m0 (a) m (b 1 ) m (c) m 2F 1 (a + m, b 2 ; c + m; z 2 ) zm 1 m!, max{ z 1, z 2 } < 1, and is represented by a single integral of Euler s type (see [14] and [15]) (4) F 1 1 B(a, c a) 1 0 t a 1 (1 t) c a 1 (1 tz 1 ) b 1 (1 tz 2 ) b 2 dt, where R(c) > R(a) > 0 and B(a, c a) is beta function [13] The generalized Hermite polynomials (known as Gould-Hopper polynomials) H r n(x, y) defined by (5) e xt+ytr n0 H r n(x, y) tn are 2-variable Kampe de Fe, riet generalization of the Hermite polynomials [1] and [4] y r x n 2r (6) H n (x, y) r!(n 2r)! [ n 2 ] r0 These polynomials usually defined by the generating function (7) e xt+yt2 n0 H n (x, y) tn, H n(x, 0) x n reduce to the ordinary Hermite polynomials H n (x) (see [13]) when y 1 and x is replaced by 2x

3 Unified (p, q)-bernoulli-hermite 127 We recall that the Hermite numbers H n are the values of the Hermite polynomials H n (x) at zero argument, that is, H n (0) A closed formula for H n is given by { 0, if n is odd, (8) H n ( 1) n/2 ( n 2 )!, if n is even 2 Preliminaries Observe that there are three independent linear transformations of Appell s function F 1 The number of z variables is diminished by one if (i) one of the b parameters equal to zero, (ii) one of the z variables equals zero, (iii) one of the z variables equals unity, or (iv) two z variables are equal One of the reduction formula out of these four cases is considered by Rahmani [12] in the form of p-bernoulli numbers B n,p by constructing an infinite matrixthe situation z 1 z 2 1 e t in (2) leads to a generating function (9) 2F 1 (1, 1; p + 2; 1 e t ) n0 B n,p t n, where 2 F 1 denotes the Gauss hypergeometric function given by (1), for the p-bernoulli numbers B n,p which are closely related to Bernoulli numbers B n by the formula (10) B n,p p + 1 p p [ p ( 1) j j j0 ] B n+j, where [ ] p j is the Stirling number of the first kind [6] Utilizing the fact that for z 1 z 2 1 e t in (2),we have a reduction formula F 1 (1, 1, q; p + 2; 1 e t, 1 e t ) 2 F 1 (1, q + 1; p + 2; 1 e t ) which is an immediate consequence of the familiar formula [15, p 54(14)] F 1 (a, a, b; c; x, x) 2 F 1 (a, b + 1; c; x) In particular,we have the following definition of the generalized p-bernoulli numbers

4 128 M A Pathan Definition 1 (p, q)-bernoulli numbers B n,p,q are given by the generating function t n (11) B n,p,q F 1(1, 1, q; p + 2; 1 e t, 1 e t ) n0 for every integer p 1 2 F 1 (1, q + 1; p + 2; 1 e t ) Equation (11) provides the equivalent definition to the Rahmani s original one given by (9) For q 0, (11) reduces to (9)Thus we have B n,p,0 B n,p and B n,0,0 B n (p, q)-bernoulli numbers B n,p,q have a series representation involving Stirling numbers of second kind [6] (12) B n,p,q ( 1) k (q + 1) { } k n k!, (p + 2) k k n0 k0 where Stirling numbers of second kind are defined by the generating function { } n t n k (et 1) k k! For q 0,(12) reduces to Kargin and Rahmani[ 7, p 2(11)] { } ( ) 1 B n,p ( 1) k n k + p + 1 k! k k k0 The first generating function for (p, q)-bernoulli numbers B n,p,q for p 1, q 0 is t n B n,1,0 2[(t 1)et + 1] (e t 1) 2 n0 and for p q 1, we have t n B n,1,1 1 t) 2(et (1 e t ) 2 2 n0 where (1 e t ) k is given by (1 e t ) k k!( 1) k Furthermore t n B n,1,2 t n B n,2,1 n0 n0 n0 { n k k0 (1 e t ) k 2 + k } t n 3 (1 e t ) 3 [(1 et )(1 + e t ) + 2te t ],

5 Since [5, eqn (21134)] Unified (p, q)-bernoulli-hermite 129 2F 1 (a, b + 1; a b; z) (1 + z) a 2F 1 (a/2, a/2 + 1/2; a b; 4z (1 + z) 2 ), the special case of (11) when a 1, p q 1 and z 1 e t yields (13) (14) t n B n, q 1,q n0 (2 et ) 1 2F 1 (1/2, 1; 1 q; 4(1 et ) (2 e t ) 2 ) From (4), we obtain an integral representation of Euler s type t n B n,p,q 1 B(1 + q, p q + 1) n0 1 0 t q (1 t) p q (1 t(1 e t )) dt where R(p q) > 1 Integral representations of Laplace type stem from the well-known results of Appell s F 1 [see, for example, 14, p 282(26) and(27)] (15) n0 Re(q) > 0 (16) B n,p,q t n 1 Γ(q) t n B n,p,q n0 0 0 e t t q 1 Φ 1 (1, 1; p + 2; 1 e t, (1 e t )t)dt, e t Φ 2 (1, q; p + 2; (1 e t )t, (1 e t )t)dt, where Φ 1 and Φ 2 are confluent forms of Appell s series defined by Humbert [14, p 25] One can notice an interesting reduction of (p, q)-bernoulli polynomials (17) 2F 1 (1, b; b + 1; 1 e t ) bζ(1 e t, 1, b), b 0, 1, 2,, where Hurwitz zeta function ζ is defined as [11] ζ(z, s, b) m0 z k (b + k) s 3 A generating function for (p, q)-bernoulli numbers Kargin and Rahmani [7] derived a generating function for the p-bernoulli numbers B n,p To this end they used the generating function of the geometric polynomials Their result is (18) 0 B n,p t n (p + 1)[(t H p)e pt (e t 1) p+1 + p k1 H k p! k!(p k) (e t 1) k+1 ],

6 130 M A Pathan where H n is the n th harmonic number defined by [6, p 258] H n It may be remarked that the above result of Kargin and Rahmani is a direct consequence of the following known result [11, p 462(128)] 2F 1 (1, n; m; z) z (n 1)! (z 1 z j1 1 j m n 1 (m 1)! (m n 1)!z { n 1 ) m n 1 [ k1 k1 (m n k 1)! (m k 1)! ( z 1 ) k 1 z z k n k + z n ln(1 z)]}, m > n Substitution of z 1 e t, n q+1 and m p+2 in the above result provide us a more general result belonging to (p, q)-bernoulli numbers considered in the preceding sectionthus we can write after a certain amount of algebra that (19) t n B n,p,q n0 p q (p + 1)! (1 e t )(p q)! [ + (et 1) q! k1 e t q ( e t 1 )p q ( k1 (p + q k)! (p + 1 k)! ( e t e t 1 )k 1 (1 e t ) k (q k + 1) + t (1 e t )], ) q+1 p > q 1 For q 0, (19) reduces to (20) n0 t n B n,p (p + 1)[ te pt p (e t 1) p+1 + e (k 1)t (p k)(e t 1) k ] k1 and for q 0 and p m 2 where m 2, 3, 4,, we have (21) t n m 1 B n,m 2 (m 1 1)e 2t [ m k ( e t e t e t 1 )k t( e t 1 )m ] n0 k2 Also a calculation shows that,for q p m 1 t n m 1 B n,m 1,m 1 m (1 e t ) m [t + (1 e t ) k ], m 1, 2, 3, k n0 k1

7 Unified (p, q)-bernoulli-hermite Unified (p, q)-bernoulli polynomials This section is devoted to (p, q)-bernoulli polynomials and their representations involving Euler polynomials E n (x) [11] Definition 2 ((p, q)-bernoulli polynomials) An explicit formula for (p,q)-bernoulli polynomials is given by the generating function (22) e xt 2F 1 (1, q + 1; p + 2; 1 e t ) for every integer p 1 For x 0 in (22),we get (23) 2F 1 (1, q + 1; p + 2; 1 e t ) n0 n0 B n,p,q (x) tn B n,p,q t n Since this reduces to Bernoulli number B n for special values of p and q, we shall call B n,p,q, a (p, q)-bernoulli number represented by the generating function (23) Since we have (24) 2F 1 (1, q + 1; p + 2; 1 e t ) e xt B n,p,q (x) tn n0 ( x) m B n,p,q (x) tn+m m! m0 n0 Therefore comparing (25) and (23),we have B n,p,q m0 Similarly, when x 1, (22) implies (25) B n,p,q ( 1) ( x) m m!(n m)! B n m,p,q(x) m0 ( 1) m B n m,p,q m!(n m)! Theorem 1 Let p 1 Then following representation for (p, q)- Bernoulli polynomials B n,p,q (x) involving Euler polynomials E n (x) holds true: (26) B n,p,q (x) 2 [ m m0 k0 B m k,p,q E n m (x) (m k)!k!(n m)! + m0 B m,p,q E n m (x) ] m!(n m)!

8 132 M A Pathan ProofThe generating function for the Euler polynomials E n (x) gives (27) e xt et n0 Substituting this value of e xt in (22) gives (28) Now using and (29) e t n0 E n (x) tn e t 2F 1 (1, q + 1; p + 2; 1 e t ) E n (x) tn 2F 1 (1, q + 1; p + 2; 1 e t ) m0 t m E n (x) tn 2F 1 (1, q + 1; p + 2; 1 e t ) t m E n (x) B m,p,q m! t n n0 n0 m0 in the left hand side of (28),we get (30) 1 2 [ m0 + t m m k0 n0 t n B m k,p,q (m k)!k! m0 n0 n0 E n (x) tn E n m (x)b m,p,q ] m k0 m0 n0 n0 B m k,p,q (m k)!k! B n,p,q (x) tn E n m (x)b m,p,q B n,p,q (x) tn Finally, replacing n by n m and comparing the coefficients of t n, we get the required result Using the following result [11, p 766] (31) E n (x) 2 n + 1 (B n+1(x) 2 n+1 B n+1 (x/2)), in (26), we have (32) Corollary 1 B n,p,q (x) m B m k,p,q { (m k)!k!(n m)! m0 k0 + B m,p,q m!(n m)! }(B n m+1(x) 2 n m+1 B n m+1 (x/2)) n m + 1

9 (33) Unified (p, q)-bernoulli-hermite 133 Setting x 0 in (32), we have the following result Corollary 2 B n,p,q m B m k,p,q { (m k)!k!(n m)! + B m,p,q m!(n m)! } m0 k0 (1 2 n m+1 )B n m+1 n m Unified (p, q)-bernoulli-hermite polynomials In this section, we define (p, q)-bernoulli-hermite polynomials and give some explicit formulas for these generalized polynomials Definition 3 ((p, q)-bernoulli-hermite polynomials) Given integer p 1 and integer q, a (p, q)-bernoulli-hermite polynomials, denoted by H Bn,p,q(x, r y) are defined by the generating function (34) e xt+ytr 2F 1 (1, q + 1; p + 2; 1 e t ) HBn,p,q(x, r y) tn n0 Using (5) and (11) in (34),we can write HBn,p,q(x, r y) tn Hn(x, r y) tn n0 n0 t m B m,p,q m! m0 which on replacing n by n m and comparing the coefficients of t n yields the representation HBn,p,q(x, r H r y) n m(x, y)b m,p,q m0 With x replaced by 2x, r 2 and y 1 it turns to the equality H n m (x)b m,p,q (35) HB n,p,q (2x, 1) To obtain (p, q)-bernoulli-hermite numbers denoted by H B n,p,q, we take in (34), r 2, y 1 and replace x by 2x Once this has been done then we take x 0 and use (8) to get H n m B m,p,q (36) HB n,p,q (0, 1) H B n,p,q m0 B m,p,q m0 m0 { 0, if n m is odd ( 1) (n m) /2 (n m)! ( n m 2 )!, if n m is even,

10 134 M A Pathan where Hermite number H n is given by (8) Theorem 2 The following summation formulae for (p, q)-bernoulli-hermite polynomials H B n,p,q holds true: (37) HBn,p,q(z, r u) m0 or, in its equivalent form (38) HBn m,p,q(x r α, y β)hm(α r x + z, β y + u), HBn,p,q(z r α + x, u β + y) HBn m,p,q(x r α, y β)hm(z, r u) m0 Proof By exploiting the generating function (34), we can write fty H Bn,p,q(z, r u) tn ezt+utr 2F 1 (1, q + 1; p + 2; 1 e t ) e (x z α)t (y u β)tr e (x α)t+(y β)tr 2F 1 (1, q + 1; p + 2; 1 e t ) e (x z α)t (y u β)tr HBn,p,q(x r α, y β) tn which readily yields n0 HBn,p,q(z, r u) tn n0 Hm(α r x + z, β y + u) tm m! m0 HBn,p,q(x r α, y β) tn, n0 and thus by replacing n by n m and comparing the coefficients of t n, we get (37) On replacing z by z α x and u by u β + y in (37), we get (38) Setting z u 0 in (37) and noting that H Bn,p,q(0, r 0) H Bn,p,q, r we have the following result for (p, q)-bernoulli polynomials B n,p,q Corollary 3 B r n,p,q m0 HBn m,p,q(x r α, y β)hm(α r x, β y) Setting z u q 0 in (37) and noting that H B r n,p,0 (0, 0) Br n,p, we have the following result for p-bernoulli polynomials

11 Corollary 4 B r n,p Unified (p, q)-bernoulli-hermite 135 m0 HBn m,p(x r α, y β)hm(α r x, β y) Theorem 3 The following summation formulae connecting Gauss hypergeometric functions 2 F 1 and (p, q)-bernoulli-hermite polynomials H Bm,p,q r holds true: (39) 2F 1 (1, p + 1 q; p + 2; 1 e t ) 2F 1 (p + 1, q + 1; p + 2; 1 e t ) 2F 1 (p + 1, p + 1 q; p + 2; 1 e t ) where P n,p,q, Q n,p,q and R n,pq are given by (40) P n,p,q (x, y) Q n,p,q (x, y) R n,p,q (x, y) n0 n0 n0 P n,pq (x, y) tn Q n,p,q (x, y) tn R n,p,q (x, y) tn, Hn m(1 r x, y) H Bm,p,q(x, r y) m0 Hn m(1 r x + q, y) H Bm,p,q(x, r y) m0 Hn m(q r x p, y) H Bm,p,q(x, r y) m0 Proof It is easy to use the linear transformations (also known as Euler s transformations) [see, S and M p 33(19), (20), (21)]of Gauss hypergeometric functions to prove that 2F 1 (1, q + 1; p + 2; 1 e t ) e t 2F 1 (1, p + 1 q; p + 2; 1 e t ) e qt t 2F 1 (p + 1, q + 1; p + 2; 1 e t ) We further rewrite these relations in the form e pt qt 2F 1 (p + 1, p + 1 q; p + 2; 1 e t ) 2F 1 (1, p + 1 q; p + 2; 1 e t ) e t xt ytr [e xt+ytr 2F 1 (1, q + 1; p + 2; 1 e t )] 2F 1 (p + 1, q + 1; p + 2; 1 e t ) e qt+t xt ytr [e xt+ytr 2F 1 (1, q + 1; p + 2; 1 e t )] 2F 1 (p + 1, p + 1 q; p + 2; 1 e t ) e pt+qt xt yt r [e xt+ytr 2F 1 (1, q + 1; p + 2; 1 e t )]

12 136 M A Pathan We may split right hand sides of these relations into product of the generating functions of the polynomials Hn(x, r y) and (p, q)-bernoulli-hermite polynomials H Bm,p,q(x, r y) Finally, replacement of n by n m and use of series arrangement technique prove the result (39) Remark 1 An important application comes from the reduction of (39) for q 0 Within such a context, we use (17) and the first two equations of (39) to get a series representation of Hurwitz zeta function (p + 1)ζ(1 e t, 1, p + 1) P n,p (x, y) tn Q n,p (x, y) tn, where (41) P n,p (x, y) Q n,p (x, y) n0 n0 Hn m(1 r x, y) H Bm,p(x, r y) m0 On the other hand,we use (11) with q p in the last equation of (39) Then we have Hn m( x, r y) H Bm,p,p(x, r y) (42) B n,p,p R n,p,p (x, y), where B n,p,p (p + 1) m0 m0 ( 1) m { m! n p m m 6 Implicit formulae involving (p, q)-bernoulli-hermite polynomials This section of the paper is devoted to employing the definition of the Hermite-Bernoulli polynomials H B n [α,m 1] (x, y) in proving the generalizations of the results of Khan et al [8], Dattoli [4, p 386(17)] and Pathan [9] (see also [10]) For the derivation of implicit formulae involving the Hermite-Bernoulli polynomials H B n [α,m 1] (x, y), the same considerations as developed for the ordinary Hermite and related polynomials in Khan et al [8]and Dattoli et al [2] and [3] hold as well First we prove the following results involving Hermite-Bernoulli polynomials H B n [α,m 1] (x, y) Theorem 4 The following summation formulae for (p, q) Bernoulli- Hermite polynomials H B r n,p,q(x, y) holds true: (43) HB r n,p,q(x, y) } B n k,p (x z)hk r (z, y) (n k)!k! k0

13 Unified (p, q)-bernoulli-hermite 137 Proof By exploiting the generating function (22), we can write equation (34) as (44) HBn,p,q(x, r y) tn e(x z)t 2F 1 (1, q + 1; p + 2; 1 e t )e zt+ytr, n0 (45) HBn,p,q(x, r y) tn B n,p (x z) tn n0 n0 k0 Hk r (z, y)tk k! Now replacing n by n k using the Lemma 3 [15, p 101(1)] in the right hand side of equation (46), we get (46) HBn,p,q(x, r y) tn n0 n0 B n k,p (x z)hk r t n (z, y) (n k)!k! k0 On equating the coefficients of the like powers of t, we get (43) Remark 2 Letting z x in (43) gives an equivalent form of (53) For p q 0 in (43), we get a known result of Pathan [10] which further for r 2 gives a known result of Dattoli [4, p 386(17)] Again taking y 0 in formula (43), we obtain (47) HB r n,p,q(x) k0 z n B n k,p (x z) (n k)!k! Theorem 5 The following summation formulae for (p, q) Bernoulli-Hermite polynomials H B r n,p,q(x, y) and (p, q) Bernoulli polynomials B r n,p,q holds true: (48) [ n r ] m0 B r n mr,p,q(x)b m,p,q (y) (n mr)!m! [ n r ] m0 HBn mr,p,q(x, r y)b m,p,q, (n mr)!m! (49) [ n r ] m0 B r n mr,p,q(x)b m,p,q (y) (n mr)!m! n k k0 m0 Hn k r (x, y)b n m k,p,qb m,p,q k!(n m k)!m! Proof Consider the definition of (p, q) Bernoulli-polynomials H B r n,p,q (50) m0 B r m,p,q(y) tmr m! eytr 2F 1 (1, q + 1; p + 2; 1 e tr ),

14 138 M A Pathan where x is replaced by y and t is replaced by t r in (22) On multiplying (22) and (49),we have (51) m0 B r m,p,q(y) tmr m! n0 B n,p,q (x) tn e xt+ytr 2F 1 (1, q + 1; p + 2; 1 e t ) 2 F 1 (1, q + 1; p + 2; 1 e tr ) HBn,p,q(x, r y) tn t mr B m,p,q m! n0 m0 Now replacing n by n rm, using the Lemma 3 [15, p 101(1)] in the right hand side of equation (46) and then equating the coefficients of the like powers of t, we get (48) Another way of defining the right hand side of (51) is suggested by replacing e xt+ytr by its series representation Bm,p,q(y) r tmr B n,p,q (x) tn m! m0 k0 n0 Hn(x, r t n y) B n,p,q n0 m0 B m,p,q t mr m! Next,we rearrange the order of summation and then equating the coefficients of the like powers of t, we get (49) Setting r 2 and y 0 in the above theorem,we have the following result for (p, q)-bernoulli polynomials (52) Corollary 5 [ n 2 ] m0 B n 2r,p,q (x)b m,p,q (n 2m)!m! n k k0 m0 H n (x)b n m k,p,q B m,p,q k!(n m k)!m! Theorem 6 The following summation formulae for (p, q)-bernoulli- Hermite polynomials H B k n,p,q(x, y) and (p, q)-bernoulli polynomials B k n,p,q holds true: (53) m0 [ n m k ] ( x y k y HB x k )r n kr m,p,q k (x, y)b m,p,q r!m!(n m kr)!y m x n m kr r0 HBn m,p,q(y, k x)b m,p,q x m y n m m0 Proof On replacing t by t/x and r by k, we can write equation (34) as (54) HBn,p,q(x, k y) tn x n t k et+y x k 2F 1 (1, q + 1; p + 2; 1 e t/x ) n0

15 Unified (p, q)-bernoulli-hermite 139 Now interchanging x and y, we have (55) HBn,p,q(y, k x) tn y n t k et+x y k 2F 1 (1, q + 1; p + 2; 1 e t/y ) n0 Comparison of (54) and (55) yields (56) e x t k tk yk y x k 2F 1 (1, q + 1; p + 2; 1 e t/y ) HBn,p,q(x, k y) tn x n n0 2 F 1 (1, q + 1; p + 2; 1 e t/x ) HBn,p,q(y, k x) tn y n n0 Using (36), we get (57) r0 ( x y k y x k ) r t kr r! m0 m0 B m,p,q B m,p,q t m x m m! t m y m HB m! n,p,q(x, k y) tn x n n0 HBn,p,q(y, k x) tn y n n0 Next, we rearrange the order of summation and then equating the coefficients of the like powers of t, we get (53) Setting p 0 and q 0 in the above theorem,we have the following result for Bernoulli polynomials Corollary 6 (58) m0 [ n m k ] ( x y k y HB x k )r n kr m k (x, y)b m r!m!(n m kr)!y m x n m kr r0 HB k n m(y, x)b m x m y n m m0 7 Concluding remarks The formulae we have provided in previous sections illustrate how properties of Appell s function F 1 can sometimes be established or understood more easily by using generalized hypergeometric series of several variablesgauss s

16 140 M A Pathan function 2 F 1 and Appell s function F 1 are both special cases of Lauricella hypergeometric function FD n [15, p 60] defined by the power series (59) FD[a, n b 1,, b n ; c; x 1,, x n ] m 1,,m n0 (a) m1 ++m n (b 1 ) m1 (b n ) mn (c) m1 ++m n z m 1 m 1! zmn n m n!, where for convergence z m < 1, m 1, 2, The following reduction of FD n will be useful (60) F n D[a, b 1,, b n ; c; x,, x] 2 F 1 (a, b b n ; c; x), because it allows one to conclude that for (p, q)-bernoulli numbers given by (11) for every p 1, we can write F n D[1, q, 1, 0, 0,, 0; p + 2; 1 e t,, 1 e t ] F n D[1, q, b 2,, b n ; c; 1 e t,, 1 e t ] 2 F 1 (1, q + 1; p + 2; 1 e t ) where b 2 + b b n 1 Our speculations for Lauricella hypergeometric function FD n [15,p60] and their reduction cases will yield perhaps a further feeling on the usefulness of the series This process of using different analytical means on their respective generating functions can be extended to drive new relations for conventional and generalized Bernoulli numbers and polynomials Acknowledgement The author would like to thank the referee for providing useful suggestions for the improvement of this paper and also the Centre for Mathematical and Statistical Sciences (CMSS) for providing facilities References [1] Bell ET, Exponential polynomials, Ann of Math, 35(1934), [2] Dattoli G, Incomplete 2D Hermite polynomials: Properties and applications, J Math Anal Appl, 284(2)(2003), [3] Dattoli G, Chiccoli C, Lorenzutta S, Maimo G, Torre A, Generalized Bessel functions and generalized Hermite polynomials, J Math Anal Appl, 178(1993), [4] Dattoli G, Lorenzutta S, Cesarano C, Finite sums and generalized forms of Bernoulli polynomials, Rendiconti di Mathematica, 19(1999), [5] Erdélyi A, Magnus W, Oberhettinger F, Tricomi FG, Higher Transcendental Functions, Vol II (Bateman Manuscript Project), McGraw- Hill, Book Co Inc, New York, Toronto and London, 1953

17 Unified (p, q)-bernoulli-hermite 141 [6] Graham RL, Knuth DE, Patashnik O, Concrete Mathematics, Addison-Wesley Publ Com, New York, 1994 [7] Kargin L,Rahmani M, A closed formula for the generating function of p-bernoulli numbers, Questiones Mathematicae, Feb (2018), 1-9, org/102989/ [8] Khan S, Pathan MA, Hassan NAM, Yasmin G, Implicit summation formula for Hermite and related polynomials, J Math Anal Appl, 344(2008), [9] Pathan MA, A new class of generalized Hermite-Bernoulli polynomials, Georgian Mathematical Journal, 19(2012), [10] Pathan MA, A new class of Tricomi-Legendre-Hermite and related polynomials, Georgian Mathematical Journal, 21(2014), [11] Prudnikov AP, Brychkov YuA, Marichev OI, Integrals and Series, Vol 3, More Special Functions, Nauka, Moscow, 1986; Translated from the Russian by GG Gould; Gordon and Breach Science Publishers, New York, Philadelphia, London, Paris, Montreux, Tokyo, Melbourne, 1990 [12] Rahmani M, On p-bernoulli numbers and polynomials, J Number Theory, 157(2015), [13] Rainville ED, Special functions, Chelsia Publishing Co, Bronx, 1971 [14] Srivastava HM, Karlsson PerW, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Ltd, Chichester, Brisbane, U K) John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985 [15] Srivastava HM, Manocha HL, A Treatise on Generating Functions, Ellis Horwood Limited, New York, 1984 M A Pathan Centre for Mathematical and Statistical Sciences (CMSS) Peechi PO Thrissur, Kerala , India mapathan@gmailcom Received on and, in revised form, on