Some symmetric identities for the generalized Bernoulli, Euler and Genocchi polynomials associated with Hermite polynomials

In 2008, Liu and Wang established various symmetric identities for Bernoulli, Euler and Genocchi polynomials. In this paper, we extend these identities in a unified and generalized form to families of Hermite–Bernoulli, Euler and Genocchi polynomials. The procedure followed is that of generating functions. Some relevant connections of the general theory developed here with the results obtained earlier by Pathan and Khan are also pointed out.

These polynomials are usually defined by the generating function as and reduce to the ordinary Hermite polynomials H n (x) (see Andrews 1985) when y = −1 and x is replaced by 2x.
(2) e xt+yt 2 = et seq.], [Luke (1969), Section 2.8] and Luo et al. 2003;Pathan 2012;Pathan and Khan 2014, 2016Simsek 2010;Srivastava et al. 2012): and It is easy to see that and Moreover For each k ∈ N 0 , S k (n) defined by is called the sum of integer powers Tuenter (2001). The exponential generating function for S k (n) is given by For each k ∈ N 0 , T k (n) defined by is called the alternating sum. The exponential generating function for T k (n) is n (x) t n n! , (|t| < π; 1 α = 1) n (x) t n n! , (|t| < π; 1 α = 1) n (x) = G (α) n (x; 1) (n ∈ N) (7) B n (1) − B n = δ (n,1) , n ≥ 0 The following are some special values where δ (i,j) is the Kronecker delta defined by δ (i,j) = 1 for i = j and δ (i,j) = 0 for i � = j. A close relation of the power sum and the Bernoulli polynomials, also the alternate sum and the Euler polynomials, can be seen in Abramowitz and Stegun [1972, Eq. (23.1.4)] as follows where n and k are nonnegative integers.
The hyperbolic cotangent and the hyperbolic tangent (Weisstein http://mathworld. wolfram.com) respectively are defined by and Therefore where Pathan and Khan (2015) introduced the generalized Hermite-Bernoulli polynomials for two variables H B (α) n (x, y) defined by which is essentially a generalization of Bernoulli numbers, Bernoulli polynomials, Hermite polynomials and Hermite-Bernoulli polynomials H B n (x, y) introduced by Dattoli et al. [1999, p. 386 (1.6)] in the form The purpose of this paper is to give some general symmetry identities for generalized Hermite-Euler, Hermite-Genocchi and mixed type polynomials by using different analytical means on their respective generating functions. These results extend some known identities of Hermite, Bernoulli, Euler, Genocchi and mixed type polynomials studied by Dattoli et al. (1999), Liu andWang (2009), Pathan (2012) and Pathan and Khan (2014, 2016.

Symmetry identities for generalized Hermite-Euler polynomials
In this section, we establish general symmetry identities for the generalized Hermite-Euler polynomials H E (α) n (x, y), of orderα and alternate power sum. Throughout this section α will be taken as an arbitrary real or complex parameter.
Theorem 1 For integers n ≥ 0, a ≥ 1 and b ≥ 1, if a and b have the same parity, then the following identity holds true: Replacing n by n − k in the R.H.S. of above equation, we get Using a similar plan, we get By comparing the coefficients of t n n! in the R.H.S of above Eqs. (21) and (22), we arrive at the desired result.
Corollary 1 For integers n ≥ 0, a ≥ 1 and b ≥ 1 , if a and bhave the same parity, then the following identity holds true: Theorem 2 For integers n ≥ 0, a ≥ 1 and b ≥ 1 , if a is odd and b is even, then the following identity holds true: is not symmetric in a and b, thus G(t) can also be expanded as Using identity (7) and comparing the coefficients of t n n! in the R.H.S. of Eqs. (25) and (26), we get the desired result.
Corollary 2 For integers n ≥ 0, a ≥ 1 and b ≥ 1 , if a is odd and b is even, then the following identity holds true: Theorem 3 For integers n ≥ 1 , a ≥ 1 and b ≥ 1, if a is even and b is odd, then the following identity holds true: Another expansion of G(t) is as follows : Now using Eq. (17) and identity (7) and comparing the coefficients of t n n! in the R.H.S. of Eqs. (29) and (30), we arrive at the desired result.
Corollary 3 For integers n ≥ 1, a ≥ 1 and b ≥ 1, if a is even and b is odd, then the following identity holds true: Theorem 4 For integers n ≥ 0, a ≥ 1 and b ≥ 1, if a and b have same parity, then the following identity holds true: Proof Let We expand G(t) as follows : Replacing n by n − k in the R.H.S. of above equation, we get Using a similar plan, we obtain By comparing the coefficients of t n n! in the R.H.S. of last two Eqs. (33) and (34), we arrive at the desired result.
Remark 4 For z = 0 in Theorem (4), the result reduces to known result of Liu and Wang [2009, Theorem 2.10].
Corollary 4 For integers n ≥ 0, a ≥ 1 and b ≥ 1, if a and b have same parity, then the following identity holds true: Theorem 5 For integers n ≥ 0, a ≥ 1 and b ≥ 1, if a is odd and b is even, then the following identity holds true: Proof Let In view of definition (1.15), G(t) has the following expansion: Using identity (7) and comparing the coefficients of t n n! in the R.H.S. of Eqs. (33) and (37), we get the result (36).
Corollary 5 For integers n ≥ 0, a ≥ 1 and b ≥ 1, if a is odd and b is even, then the following identity holds true: l=0,l� =n n + 1 l B n+1−l n + 1 b n−l l k=0 l k a n−k b k

Symmetric identities for Hermite-Genocchi polynomials
In this section, we derive some symmetry identities for Hermite-Genocchi polynomials H G n (x, y). We now begin the following theorems.
Theorem 6 For integers n ≥ 0, a ≥ 1 and b ≥ 1, if a and Now following (41) we expand P(t) as follows: On the similar lines, we can show that Comparing the coefficients of t n n! in the R.H.S. of above Eqs. (44) and (45), yields identity (40).

.2) and (4.3)].
Corollary 6 For integers n ≥ 0, a ≥ 1 and b ≥ 1, if a and b have the same parity, then the following identity holds true: and Theorem 7 For integers n ≥ 0, a ≥ 1 and b ≥ 1, if a is odd and b is even, then the following identity holds true: (44) We can expand P(t) as follows:

Another expansion of P(t) is
Comparing the coefficients of t n n! in Eqs. (43) and (50), we obtain the first identity. Also comparing the coefficients of t n n! in the R.H.S. of Eqs. (44) and (51), we obtain the second identity.
Corollary 7 For integers n ≥ 0, a ≥ 1 and b ≥ 1, if a is odd and b is even, then the following identity holds true: and Theorem 8 For integers n ≥ 0, a ≥ 1 and b ≥ 1, if a is even and b is odd, then the following identity holds true: and Proof Let we expand P(t) as follows: Comparing the coefficients of t n n! in the R.H.S. of Eqs. (42) and (57), we have Finally using [Liu and Wang (2009), p. 3348, (2.4)], we arrive at the desired result (54).
Corollary 8 For integers n ≥ 0 , a ≥ 1 and b ≥ 1, if a is even and b is odd, then the following identity holds true: and

Mixed type identities
In this section, we establish some mixed type identities involving the generalized Hermite-Bernoulli and Euler polynomials of order α. Throughout this section α will be taken as an arbitrary real or complex parameter.
Theorem 9 For integers n ≥ 1, a ≥ 1 and b ≥ 1, if a is even, then the following identity holds true: and Proof Let Since a is even, we also have Comparing coefficients of t n n! in the R.H.S. of above Eqs. (65) and (66), yields the first identity (62).
Again, we expand H(t) as follows: On similar lines, we can expand H(t) as follows: Comparing the coefficients of t n n! in the R.H.S of above Eqs. (67) and (68), we get the result (63).

Corollary 9
For integers n ≥ 1, a ≥ 1 and b ≥ 1, if a is even, then the following identity holds true: and Theorem 10 For integers n ≥ 1, a ≥ 1 and b ≥ 1, if a is odd, then the following identity holds true: On the other hand Comparing the coefficients of t n n! in the R.H.S of above Eqs. (78) and (79), yields the desired result.

Corollary 11
For integers n ≥ 1, a ≥ 1 and b ≥ 1, the following identity holds true: Theorem 12 For integers n ≥ 1, a ≥ 1 and b ≥ 1, the following identity holds true: Proof Let