Palestine Journal of Mathematics

Vol. 7(1)(2018) , 35–52 © Palestine Polytechnic University-PPU 2018

SOME IDENTITIES OF THE APOSTOL TYPE POLYNOMIALS ARISING FROM Ghazala Yasmin

Communicated by P. K. Banerji

MSC 2010 Classications: 33C45, 33C50, 33E20.

Keywords and phrases: Apostol type polynomials; Appell and Sheffer sequences; Associated sequences; Umbral calcu- lus.

This work has been done under UGC-BSR Reaserch Start-Up-Grant (Ofce Memo No. 30-90/2015(BSR)) awarded to the author by the University Grants Commission (UGC), Government of India, New Delhi. Abstract The aim of this paper is to introduce and investigate several new identities related to the unied families of Apostol type polynomials. The results presented here are based upon the theory of the umbral calculus and the umbral algebra. We consider Apostol type polynomials related to associated sequences of polynomials and nally give some new and interesting iden- tities of those polynomials arising from transfer formula for the associated sequences. Further, we derive several identities involving Apostol type polynomials arising from umbral calculus to have alternative ways. Some new and known identities are also derived for the families of Apostol type polynomials using umbral calculus as special cases.

1 Introduction

Umbral calculus provide powerful tool to deal with the properties of special polynomials and functions. It has been used in numerous problems of mathematics and its related eld like com- binotorics (for example, see [1, 3, 6, 7, 9]). Its techniques have been used in different areas of physics for example, it was used in group theory and quantum mechanics by Biedenharn et al. [5, 6]. Here we rst recall the notations and denitions related to the umbral algebra and calculus [32, 33].

Let F be the set of all formal power series in the variable t over C with ( ) α F = f(t) = S∞ k tk|α ∈ C . k=0 k! k Let us assume that P be the algebra of polynomials in the variable x over C and P∗ is the vector space of all linear functionals on P. As a notation, the action of the linear functional L on a polynomial p(x) is denoted by ⟨L|p(x)⟩. The formal power series α f(t) = S∞ k tk ∈ F, k=0 k! denes a linear functional on P by setting

n ⟨f(t)|x ⟩ = αn, (n ≥ 0). (1.1) From (1.1), we note that k n ⟨t |x ⟩ = n!δn,k, (n, k ≥ 0), (1.2) where δn,k is the Kronecker symbol (see [10, 14, 15, 32, 33]).

k Let S∞ ⟨L|x ⟩ k. Then, by (1.1), we get ⟨ | n⟩ ⟨ | n⟩ . The map → fL(t) = k=0 k! t fL(t) x = L x L fL(t) is a vector space isomorphism from P∗ onto F. Hence forth, F thought of as both a formal power series and a linear functional. We shall call F the umbral algebra. The umbral calculus 36 Ghazala Yasmin is the study of umbral algebra (see [10, 14, 15, 31, 32, 33]).

The order o(f(t)) of the non-zero power series f(t) is the smallest integer k for which the coefcient of tk does not vanish (see [10, 14, 15, 32, 33]). If o(f(t)) = 1, then f(t) is called a delta series, and if o(f(t)) = 0, then f(t) is called an invertible series. Let o(f(t)) = 1 and o(g(t)) = 0. Then there exists a unique sequence Sn(x) of polynomials such that k ⟨g(t)f(t) |Sn(x)⟩ = n!δn,k, (n, k ≥ 0). The sequence Sn(x) is called Sheffer sequence for (g(t), f(t)), which is denoted by Sn(x) ∼ (g(t), f(t)). If Sn(x) ∼ (1, f(t)), then Sn(x) is called the associated sequence for f(t) (see [10, 14, 31, 32, 33]).

From (1.1), we note that ⟨eyt|p(x)⟩ = p(y). Let f(t) ∈ F and p(x) ∈ P. Then we have

⟨f(t)|xk⟩ ⟨tk|p(x)⟩ f(t) = S∞ tk, p(x) = S∞ xk (1.3) k=0 k! k=0 k! and by (1.2), we get:

p(k)(0) = ⟨tk|p(x)⟩⟨1|p(k)(x)⟩ = p(k)(0). (1.4) Thus from (1.3), we have

dK p(x) tkp(x) = p(k)(x) = . (1.5) dxk Also we note that ⟨f(t)(g(t)|p(x)⟩ = ⟨(g(t)|f(t)p(x)⟩.

Let Sn(x) ∼ (g(t), f(t)). Then we see that 1 yf¯(t) ∞ Sk(y) k e = S 0 t , ∀ y ∈ C, (1.6) g(f¯(t)) k= k! ¯ where f(t) is the compositional inverse of f(t) (see [32, 33]). Let pn(x) ∼ (1, f(t)), qn(x) ∼ (1, g(t)). Then, the transfer formula for the associated sequence is given by ( ) f(t) n q (x) = x x−1p (x). (1.7) n g(t) n

Further, let Sn(x) ∼ (g(t), f(t)) and rn(x) ∼ (h(t), l(t)) then Sn 1 8 Sn(x) = k=0Cn,krk(x), ( . ) where the connection constant Cn,k are given by 1 ¯ h(f(t)) ¯ k n Cn,k = ⟨ l(f(t)) |x ⟩. (1.9) k! g(f¯(t)) Equations (1.8) and (1.9) are called the alternative ways of Sheffer sequences.

(α) (α) (α) The polynomials Bn (x), En (x) and Gn (x) are dened by the following generating func- tions [11, 34]: ( ) ∞ t α ∑ tn ext = B(α)(x) , |t| < 2π, (1.10) et − 1 n n! n=0 ( ) ∞ 2 α ∑ tn ext = E(α)(x) , |t| < π, (1.11) et + 1 n n! n=0 ( ) ∞ 2t α ∑ tn ext = G(α)(x) , |t| < π. (1.12) et + 1 n n! n=0 SOME IDENTITIES OF THE APOSTOL TYPE POLYNOMIALS ... 37

It is easy to see that Bn(x), En(x) and Gn(x) are given, respectively, by

(1) ; (1) ; (1) ∈ N : N ∪ {0} 1 13 Bn (x) = Bn(x) En (x) = En(x) Gn (x) = Gn(x), n 0 = . ( . ) Some interesting analogues of the classical Bernoulli and Euler polynomials were rst inves- tigated by Apostol [2] and further studied by Srivastava [36]. Luo and Srivastava [27] introduced (α) the Apostol Bernoulli polynomials of order α ∈ N0, denoted by Bn (x; λ), λ ∈ C, which are dened by the

( ) ∞ t α ∑ tn ext = B(α)(x; λ) , |t| < 2π, when λ = 1; |t| < |logλ|, when λ ≠ 1, (1.14) λet − 1 n n! n=0 with (α) ; 1 (α) 1 15 Bn (x ) = Bn (x). ( . ) (α) The Apostol Euler polynomials of order α ∈ N0, denoted by En (x; λ), λ ∈ C are introduced by Luo [21] and are dened by the generating function

( ) ∞ 2 α ∑ tn ext = E(α)(x; λ) , |t| < |log(−λ)|, (1.16) λet + 1 n n! n=0 with (α) ; 1 (α) 1 17 En (x ) = En (x). ( . )

Further, Luo [24] introduced the Apostol Genocchi polynomials of order α ∈ N0, denoted by (α) Gn (x; λ), λ ∈ C, which are dened by the generating function ( ) ∞ 2t α ∑ tn ext = G(α)(x; λ) , |t| < |log(−λ)|, (1.18) λet + 1 n n! n=0 with G(α) ; 1 (α) 1 19 n (x ) = Gn (x). ( . ) Certain properties of the Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomi- als such as asymptotic estimates, fourier expansions, multiplication formulas etc. are studied by several researchers, see for example [4, 17, 22, 23, 25, 30]. Recently, Luo and Srivastava [28] (α) introduced the Apostol-type polynomials Fn (x; λ; µ, ν)(α ∈ N, λ, µ, ν ∈ C) of order α, which are dened by means of the generating function

( ) ∞ 2µ tν α ∑ tn ext = F (α)(x; λ; µ, ν) , |t| < |log(−λ)| (1.20) λet + 1 n n! n=0 where F (α) ; : F (α) 0; ; 1 21 n (λ µ, ν) = n ( λ µ, ν), ( . ) denotes the Apostol type numbers of order α, dened by the generating function

( ) ∞ 2µ tν α ∑ tn = F (α)(λ; µ, ν) , |t| < |log(−λ)|. (1.22) λet + 1 n n! n=0

(α) These polynomials are viewed as a unication and generalization of the polynomials Bn (x; λ), (α) (α) En (x; λ) and Gn (x; λ). In fact, from equations (1.14), (1.16), (1.18) and (1.20), we have

−1 αF (α) ; − ; 0 1 (α) ; 1 23 ( ) n (x λ , ) = Bn (x λ), ( . )

F (α) ; ; 1 0 (α) ; 1 24 n (x λ , ) = En (x λ), ( . ) F (α) ; ; 1 1 G(α) ; 1 25 n (x λ , ) = n (x λ). ( . ) 38 Ghazala Yasmin

Recently, some results for the Apostol type polynomials and their generalized forms are es- tablished, see for example [8, 18, 19, 20, 26, 35].

The Hermite polynomials [11] are dened by the generating function

∑∞ n 2 t e2xt−t = H (x) . (1.26) n n! n=0 where

Hn = Hn(0) denotes the Hermite numbers.

The Touchard polynomials [29] are dened by the generating function

∑ n t t ex(e −1) = T (x) . (1.27) n n! n≥0 By comparing (1.27) with the generating function of the Bell numbers given by

∑ n t t ee −1 = Bell , (1.28) n n! n≥0 we get the following relationship between the Touchard polynomial and the Bell numbers

Tn(1) = Belln. (1.29) Furthermore, we note that the rst is given by

∑n l (x)n = x(x − 1)(x − n + 1) = S1(n, l) x , (see [14, 32, 33]) (1.30) l=0 and the second Stirling number is dened by the generating function

∑∞ l t n t (e − 1) = n! S2(l, n) , (see [10, 32, 33]). (1.31) l! l=n Motivated by the above mentioned work on Apostol type polynomials and due to the im- portance of the umbral calculus in this paper, we consider Apostol type polynomials related to associated sequences of polynomials by the use of umbral calculus. Finally, we give some new and interesting identities of those polynomials arising from transfer formula for the associated sequences. Further, we establish a connection between our polynomial and several known fami- lies of polynomials arising from umbral calculus to have alternative ways.

2 Umbral calculus and Apostol type polynomials

From (1.6), we note that (( ) ) λez + 1 α F (α)(x; λ; µ, ν) ∼ , t . (2.1) n 2µ tν Thus, we get ( ) 2µ zν α F (α)(x; λ; µ, ν) = xn. (2.2) n λez + 1 Let us assume that ( ( ) ) λez + 1 α p (x) ∼ (1, z(λez + 1)), q (x) ∼ 1, z , (α ≠ 0). (2.3) n n 2µ zν From xn ∼ (1, z), (1.7) and (2.3), we note that ( ) 1 n x p (x) = x xn−1 = F (n) (x; λ; µ, ν) (2.4) n λez + 1 (2µ zν )n n−1 SOME IDENTITIES OF THE APOSTOL TYPE POLYNOMIALS ... 39 and ( ) 2µ zν αn q (x) = x xn−1 = xF (αn)(x; λ; µ, ν). (2.5) n λez + 1 n−1 In view of the fact (1.7), (2.3), (2.4) and (2.5), we can derive

( )α ( λez +1 ) x 2µ zν n F (n) (x; λ; µ, ν) = x x−1x F (αn)(x; λ; µ, ν) (2µ zν )n n−1 λez + 1 n−1

(α−1)n ( ) x ∑ (α − 1)n = λlF (αn)(x + l; λ; µ, ν), (2.6) (2µ zν )αn l n−1 l=0 where α, n ∈ N . Therefore, by (2.6), we obtain the following theorem.

Theorem 2.1. For α, n ∈ N, we have :

−1 ( ) 1 (α∑)n − 1 F (n) ; ; (α )n lF (αn) ; ; 2 7 n−1(x λ µ, ν) = −1 λ n−1 (x + l λ µ, ν). ( . ) 2µ ν (α )n l ( z ) l=0 Further in view of relation (1.23), we deduce the following result from (2.7)

Corollary 2.2. For α, n ∈ N, we have :

(α−1)n ( ) 1 ∑ (α − 1)n B(n) (x; λ) = (−λ)lB(αn)(x + l; λ), (2.8) n−1 (α−1)n l n−1 (z) l=0 Also in view of relation (1.24), we deduce the following result from (2.7)

Corollary 2.3. For α, n ∈ N, we have :

(α−1)n ( ) 1 ∑ (α − 1)n E(n) (x; λ) = λlE(αn)(x + l; λ), (2.9) n−1 2 (α−1)n l n−1 ( ) l=0 Furthermore, in view of relation (1.25), we deduce the following result from (2.7)

Corollary 2.4. For α, n ∈ N, we have :

(α−1)n ( ) 1 ∑ (α − 1)n G(n) (x; λ) = λlG(αn)(x + l; λ). (2.10) n−1 2 (α−1)n l n−1 ( z) l=0

Further, let us consider the following associated sequences: ( ( ) ) x 2µ zν α F (n) (x; λ; µ, ν) ∼ (1, z(λez + 1)), p (x) ∼ 1, z , (α ≠ 0). (2.11) (2µ zν )n n−1 n λez + 1

For xn ∼ (1, z), by (1.7) and (2.11), we get ( ) ( ) n z 1 αn z −1 λe + n−1 pn(x) = x ( )α x x = x x 2µ zν 2µ zν z λez +1 ( ) x ∑αn αn = λl(x + l)n−1. (2.12) (2µ zν )αn l l=0 40 Ghazala Yasmin

For n ≥ 1, by (1.7) and (2.11), we get ( ) z λez 1 n x ( +)) −1 F (n) ; ; pn(x) = x α x n−1(x λ µ, ν) 2µ zν (2µ zν )n z λez +1 1 = x (λez + 1)(α+1)nF (n) (x; λ; µ, ν). (2.13) (2µ zν )(α+1)n n−1 By (2.12) and (2.13), we get ( ) ∑αn αn 1 λl(x + l)n−1 = (λez + 1)(α+1)nF (n) (x; λ; µ, ν). (2.14) l (2µ zν )n n−1 l=0 Therefore, by (2.14), we obtain the following theorem.

∪ Theorem 2.5. For n ≥ 1 and α ∈ Z+ = N {0}, we have :

( ) (α+1)n ( ) ∑αn αn 1 ∑ (α + 1)n (x + l)n−1 = F (n) (x + l; λ; µ, ν). (2.15) l (2µ zν )n l n−1 l=0 l=0 Further in view of relation (1.23), we deduce the following result from (2.15)

∪ Corollary 2.6. For n ≥ 1 and α ∈ Z+ = N {0}, we have :

( ) (α+1)n ( ) ∑αn αn 1 ∑ (α + 1)n (x + l)n−1 = B(n) (x + l; λ). (2.16) l (z)n l n−1 l=0 l=0 Also in view of relation (1.24), we deduce the following result from (2.15)

∪ Corollary 2.7. For n ≥ 1 and α ∈ Z+ = N {0}, we have :

( ) (α+1)n ( ) ∑αn αn 1 ∑ (α + 1)n (x + l)n−1 = E(n) (x + l; λ). (2.17) l (2)n l n−1 l=0 l=0 Furthermore, in view of relation (1.25), we deduce the following result from (2.15)

∪ Corollary 2.8. For n ≥ 1 and α ∈ Z+ = N {0}, we have :

( ) (α+1)n ( ) ∑αn αn 1 ∑ (α + 1)n (x + l)n−1 = G(n) (x + l; λ). (2.18) l (2 z)n l n−1 l=0 l=0 Let us consider the following associated sequences: ( ( ) ) λez + 1 α (x) ∼ (1, ez − 1), xF (αn)(x; λ) ∼ 1, z , (α ≠ 0), (2.19) n n−1 2µ zν By (1.7) and (2.19), we get ( ) ez − 1 n F (αn) ; ( ) −1 2 20 x n−1 (x λ) = x α x (x)n, ( . ) λez +1 z 2µ zν ( ) ( ) ez − 1 n 2µ zν αn = x (x − 1) −1. z λez + 1 n SOME IDENTITIES OF THE APOSTOL TYPE POLYNOMIALS ... 41

Replacing x by x + 1, we have ( z )n( µ ν )αn (αn) e − 1 2 z (x + 1)F (x + 1; λ) = (x + 1) (x) −1, n−1 z λez + 1 n or

( ) ( ) n−1 z − 1 n 2µ ν αn ∑ (αn) e z l F (x + 1; λ) = S1(n − 1, l)x , n−1 z λez + 1 l=0 n∑−1 ∑l ! n (αn) k = S1(n − 1, l)S2(k + n, n) (l) F (x, λ; µ; ν)z . (2.21) (k + n)! k l−k l=0 k=0 Therefore by (2.21), we obtain the following theorem

Theorem 2.9. For n ≥ 1 and α ∈ Z+, we have : n∑−1 ∑l l (αn) (k) (αn) F (x + 1; λ) = S1(n − 1, l)S2(k + n, n) F (x, λ; µ; ν). (2.22) n−1 (k+n) l−k l=0 k=0 ( n ) Further in view of relation (1.23), we deduce the following result from (2.22)

Corollary 2.10. For n ≥ 1 and α ∈ Z+, we have : n∑−1 ∑l l (αn) (k) (αn) B (x + 1; λ) = S1(n − 1, l)S2(k + n, n) B (x, λ). (2.23) n−1 (k+n) l−k l=0 k=0 ( n ) Also in view of relation (1.24), we deduce the following result from (2.22)

Corollary 2.11. For n ≥ 1 and α ∈ Z+, we have : n∑−1 ∑l l (αn) (k) (αn) E (x + 1; λ) = S1(n − 1, l)S2(k + n, n) E (x, λ). (2.24) n−1 (k+n) l−k l=0 k=0 ( n ) Furthermore, in view of relation (1.25), we deduce the following result from (2.22)

Corollary 2.12. For n ≥ 1 and α ∈ Z+, we have : n∑−1 ∑l l (αn) (k) (αn) G (x + 1; λ) = S1(n − 1, l)S2(k + n, n) G (x, λ). (2.25) n−1 (k+n) l−k l=0 k=0 ( n ) Let ( ( ) ) λez + 1 α xF (αn)(x; λ; µ; ν) ∼ 1, z , (α ≠ 0) n−1 2µ zν z (x)n ∼ (1, e − 1), (2.26) Then we have ( )α ( λez +1 ) z 2µ zν n (x) = x x−1 x F (αn)(x; λ; µ; ν) n ez − 1 n−1 ( ) ( ) z n λez + 1 αn = x F (αn)(x; λ; µ; ν), (2.27) ez − 1 2µ zν n−1 (n) = xBn−1(x). and ∑n n∑−1 l l (x)n = S1(n, l)x = x S1(n, l + 1)x , (n ≥ 1). (2.28) l=0 l=0 Therefore by (2.27) and (2.28), we get the following theorem 42 Ghazala Yasmin

Theorem 2.13. For n ≥ 1 and 0 ≤ l ≤ n − 1, we have : ( ) n − 1 (n) S1(n, l + 1) = B . (2.29) l n−l−1 Also from (2.27), we note that ( ) z − 1 n e z αn µ ν −αn (αn) (x − 1) −1 = (λe + 1) (2 z ) F (x; λ; µ; ν), (n ≥ 1). (2.30) z n n−1 L.H.S. of (2.30)

( ) n−1 z − 1 n ∑ e l = S1(n − 1, l)(x − 1) z l=0

n∑−1 ∑l l (k) l−k = S1(n − 1, l) S2(k + n, n)(x − 1) . (2.31) (k+n) l=0 k=0 ( n ) and R.H..S of (2.30). ( ) ∑αn αn = (2µ zν )−αn λlF (αn)(x + l; λ; µ; ν) . (2.32) l n−1 l=0 Therefore, by (2.30), (2.31) and (2.32), we obtain the following theorem.

Theorem 2.14. For n ≥ 1 and α ∈ Z+, we have : ( ) ∑αn n∑−1 ∑l l µ ν −αn αn l (αn) (k) l−k (2 z ) λ F (x+l; λ; µ; ν) = S1(n−1, l)S2(k+n, n)(x−1) l n−1 (k+n) l=0 l=0 k=0 ( n ) (2.33) Further in view of relation (1.23), we deduce the following result from (2.33)

Corollary 2.15. For n ≥ 1 and α ∈ Z+, we have : ( ) ∑αn n∑−1 ∑l l −αn αn l (αn) (k) l−k z (−λ) B (x+l; λ) = S1(n−1, l)S2(k+n, n)(x−1) (2.34) l n−1 (k+n) l=0 l=0 k=0 ( n ) Also in view of relation (1.24), we deduce the following result from (2.33)

Corollary 2.16. For n ≥ 1 and α ∈ Z+, we have : ( ) ∑αn n∑−1 ∑l l −αn αn l (αn) (k) l−k 2 λ E (x + l; λ) = S1(n − 1, l)S2(k + n, n)(x − 1) (2.35) l n−1 (k+n) l=0 l=0 k=0 ( n ) Furthermore, in view of relation (1.25), we deduce the following result from (2.33)

Corollary 2.17. For n ≥ 1 and α ∈ Z+, we have : ( ) ∑αn n∑−1 ∑l l −αn αn l (αn) (k) l−k (2z) (λ) G (x + l; λ) = S1(n − 1, l)S2(k + n, n)(x − 1) l n−1 (k+n) l=0 l=0 k=0 ( n ) (2.36) SOME IDENTITIES OF THE APOSTOL TYPE POLYNOMIALS ... 43

Let ( ( ) ) 2µ zν α p (x) ∼ 1, z , (x) ∼ (1, ez − 1), (α ≠ 0). (2.37) n λez + 1 n By (2.12), we have ( ) x ∑αn αn p (x) = (x + l)n−1 n (2µ zν )αn l l=0

( ) −1 ( ) 1 ∑αn αn n∑ n − 1 = x kn−1−l(x)1. (2.38) (2µ zν )αn k l k=0 l=0 From (1.7) and (2.37), we have

( )α ( λez +1 ) z 2µ zν n (x) = x x−1p (x) n ez − 1 n

( ) ( ) ( ) αn n−1 ( )( ) z n λez + 1 αn 1 αn ∑ ∑ αn n − 1 = x kn−1−l(x)1 λez + 1 2µ zν 2µ zν k l k=0 l=0 ( ) n−1 ( αn n−1 l ( )( )( )( ) ) 1 αn ∑ ∑ ∑ ∑ αn n − 1 l m = x kn−1−lF (αn)(x; λ; µ; ν)B(n) xp. 2µ zν k l m p l−m m−p p=0 k=0 l=p m=p (2.39) Therefore, by (2.28) and (2.39), we obtain the following theorem.

Theorem 2.18. For n ≥ 1, α ∈ Z+ and 0 ≤ l ≤ n − 1 we have :

( ) αn n−1 l ( )( )( )( ) 1 αn ∑ ∑ ∑ − 1 αn n l m n−1−l (αn) (n) S1(n, p+1) = k F (x; λ; µ; ν)B . 2µ zν k l m p l−m m−p k=0 l=p m=p (2.40) Further in view of relation (1.23), we deduce the following result from (2.40)

Corollary 2.19. For n ≥ 1, α ∈ Z+ and 0 ≤ l ≤ n − 1 we have :

( ) αn n−1 l ( )( )( )( ) 1 αn ∑ ∑ ∑ − 1 αn n l m n−1−l (αn) (n) S1(n, p+1) = k B (x; λ)B . (2.41) z k l m p l−m m−p k=0 l=p m=p Also in view of relation (1.24), we deduce the following result from (2.40)

Corollary 2.20. For n ≥ 1, α ∈ Z+ and 0 ≤ l ≤ n − 1 we have :

( ) αn n−1 l ( )( )( )( ) 1 αn ∑ ∑ ∑ − 1 αn n l m n−1−l (αn) (n) S1(n, p+1) = k E (x; λ)B . (2.42) 2 k l m p l−m m−p k=0 l=p m=p Furthermore, in view of relation (1.25), we deduce the following result from (2.40)

Corollary 2.21. For n ≥ 1, α ∈ Z+ and 0 ≤ l ≤ n − 1 we have :

( ) αn n−1 l ( )( )( )( ) 1 αn ∑ ∑ ∑ − 1 αn n l m n−1−l (αn) (n) S1(n, p + 1) = k G (x; λ)B . 2 z k l m p l−m m−p k=0 l=p m=p (2.43) In the next section, we use alternating ways to get new identities for the Apostol type poly- nomials. 44 Ghazala Yasmin

3 Connections with families of polynomials

In this section, we present several identities involving the Apostol type polynomials and some other families of polynomials. In order to established these identities we use umbral calculus to have alternative ways. For instance, we obtain the connection of Apostol type polynomials with Hermite and Touchard polynomials.

3.1 Connections to the Hermite polynomials Let us consider the following Sheffer sequences: (( ) ) z α 2 λe + 1 z z F (α)(x; λ; µ; ν) ∼ , z ,H (x) ∼ (e 4 , ). (3.1.1) n 2µ zν n 2 Then, by (1.8), we assume that

∑n F (α) ; ; ; 3 1 2 n (x λ µ ν) = Cn,kHk(x). ( . . ) k=0 From (1.9) and (3.1.2), we have

2 z ( ) 1 4 k e z n Cn,k = ⟨( )α |x ⟩ k! λez +1 2 2µ zν ( ) 2 µ ν α 1 z 2 z = ⟨e 4 | zkxn⟩ k!2k λez + 1 ( ) n−k ∑2 1 −k n (α) = 2 (n − k)2 F (0; λ; µ; ν). (3.1.3) k 22ll! l n−k−2l l=0 Therefore, by (3.1.2) and (3.1.3), we obtain the following theorem

Theorem 3.1. For n ≥ 0, we have :

∑n ∑ F (α) ; ; (α) n−k−l(λ µ ν) F (x; λ; µ; ν) = n! Hk(x). (3.1.4) n k!(n − k − l)!2k+l( l )! k=0 0≤l≤n−k,l:even 2 Further in view of relation (1.23), we deduce the following result from (3.1.4)

Corollary 3.2. For n ≥ 0, we have :

∑n ∑ B(α) (α) n−k−l(λ) B (x; λ) = n! Hk(x). (3.1.5) n k!(n − k − l)!2k+l( l )! k=0 0≤l≤n−k,l:even 2 which for λ = 1 gives the result of [16; Theo. 2.2].

Also in view of relation (1.24), we deduce the following result from (3.1.4)

Corollary 3.3. For n ≥ 0, we have :

∑n ∑ E(α) (α) n−k−l(λ) E (x; λ) = n! Hk(x). (3.1.6) n k!(n − k − l)!2k+l( l )! k=0 0≤l≤n−k,l:even 2 which for λ = 1 gives the result of [16; Theo. 2.1]. SOME IDENTITIES OF THE APOSTOL TYPE POLYNOMIALS ... 45

Furthermore, in view of relation (1.25), we deduce the following result from (3.1.4)

Corollary 3.4. For n ≥ 0, we have :

∑n ∑ F (α) (α) n−k−l(λ) G (x; λ) = n! Hk(x). (3.1.7) n k!(n − k − l)!2k+l( l )! k=0 0≤l≤n−k,l:even 2 Further, note that 2 z z ∼ 4 Hn(x) (e , 2). Thus we have 2 z z z 4 ∼ 1 2 n ∼ 1 3 1 8 e Hn(x) ( , 2) and ( x) ( , 2). ( . . ) From (3.1.8), we have

2 2 z z 4 n − 4 n e Hn(x) = (2x) ⇐⇒ Hn(x) = e (2x) . (3.1.9) Also let us assume that

∑n F (α) ; ; ; 3 1 10 Hn(x) = Cn,k k (x λ µ ν). ( . . ) k=0 From (1.9), (3.1.9) and (3.1.10), we get

( )α λe2z +1 1 2µ+ν ν ⟨ z 2 k| n⟩ Cn,k = 2 ( z) x ! (2z) k e 4 ( ) 1 λez + 1 α (z)2 = ⟨ (z)k|e− (2x)n⟩ k! 2µ zν 4 1 C = ⟨(λez + 1)α(z)k−νr|H (x)⟩ n,k 2µrk! n ( ) ( ) 1 ∑r n k−νr r j jz = 2 λ ⟨e |H − (x)⟩. (3.1.11) 2µr k j n k+νr j=0 Therefore, by (3.1.10) and (3.1.11), we obtain the following theorem

Theorem 3.5. For n ≥ 0, we have :

( ) ( ) 1 ∑n ∑r n k r j (α) H (x) = 2 λ H − (j)F (x; λ; µ; ν). (3.1.12) n 2(µ+ν)r k j n k+νr k k=0 j=0 Further in view of relation (1.23), we deduce the following result from (3.1.12)

Corollary 3.6. For n ≥ 0, we have : ( ) ( ) 1 ∑n ∑r n k r j (α) H (x) = 2 λ H − (j)B (x; λ). (3.1.13) n 2r k j n k+r k k=0 j=0 Also in view of relation (1.24), we deduce the following result from (3.1.12) 46 Ghazala Yasmin

Corollary 3.7. For n ≥ 0, we have : ( ) ( ) 1 ∑n ∑r n k r j (α) H (x) = 2 λ H − (j)E (x; λ). (3.1.14) n 2r k j n k k k=0 j=0 which for λ = 1 gives the result of [16; Theo. 2.5].

Furthermore, in view of relation (1.25), we deduce the following result from (3.1.12)

Corollary 3.8. For n ≥ 0, we have : ( ) ( ) 1 ∑n ∑r n k r j (α) H (x) = 2 λ H − (j)G (x; λ). (3.1.15) n 22r k j n k+r k k=0 j=0 Again let us assume that

∑n F (α) ; ; ; 3 1 16 Hn(x) = Cn,k k (x λ µ ν). ( . . ) k=0 From (1.9), (3.1.9) and (3.1.16), we get

( )α λe2z +1 1 2µ+ν ν ⟨ z 2 k| n⟩ Cn,k = 2 ( z) x ! (2z) k e 4 ( ) 1 λez + 1 α (z)2 = 2n⟨ e− 4 (z)kxn⟩ k! 2µ zν | ( ) ∞ n ∑ (−1)l z2l C = 2n−r ⟨(λez + 1)α| xn−k−r⟩ n,k k − r l!22l l=0

n−k−r ( )( ) 1 ∑r ∑2 n r (−1)l(n − k + r)! = 2k−r λj|(2j)n−k+r−2l. (3.1.17) 2r k − r j l!(n − k + r − 2l)! j=0 l=0 Therefore, by (3.1.16) and (3.1.17), we obtain the following theorem

Theorem 3.9. For n ≥ 0, we have :

n−k−r ( )( ) 1 ∑n ∑r ∑2 n r (−1)l(n − k + r)! H (x) = 2k−r λj(2j)n−k+r−2lF (α)(x; λ; µ; ν). n 2r k − r j l!(n − k + r − 2l)! k k=0 j=0 l=0 (3.1.18) Further in view of relation (1.23), we deduce the following result from (3.1.18)

Corollary 3.10. For n ≥ 0, we have :

n−k−r ( )( ) 1 ∑n ∑r ∑2 n r (−1)l(n − k + r)! H (x) = 2k−r (−λ)j(2j)n−k+r−2lB(α)(x; λ). n 2r k − r j l!(n − k + r − 2l)! k k=0 j=0 l=0 (3.1.19) Also in view of relation (1.24), we deduce the following result from (3.1.18) SOME IDENTITIES OF THE APOSTOL TYPE POLYNOMIALS ... 47

Corollary 3.11. For n ≥ 0, we have :

n−k−r ( )( ) 1 ∑r ∑2 n r (−1)l(n − k + r)! H (x) = 2k−r λj(2j)n−k+r−2lE(α)(x; λ). n 2r k − r j l!(n − k + r − 2l)! k j=0 l=0 (3.1.20) Furthermore, in view of relation (1.25), we deduce the following result from (3.1.18)

Corollary 3.12. For n ≥ 0, we have :

n−k−r ( )( ) 1 ∑r ∑2 n r (−1)l(n − k + r)! H (x) = 2k−r λj(2j)n−k+r−2lG(α)(x; λ). n 2r k − r j l!(n − k + r − 2l)! k j=0 l=0 (3.1.21)

3.2 Connections to the Touchard polynomials Using Similar technique used in previous theorems, we can express our 2-variable Apostol type polynomials in terms of other families. For instance we can obtain Apostol type polynomials in terms of Touchard polynomials Tn(x) Using the facts that Tn(x) ∼ (1, log(t + 1)) and

∑n F (α) ; ; ; 3 2 1 n (x λ µ ν) = Cn,kTk(x). ( . . ) k=0 with (2.1) and (1.9) we obtain the following result

Theorem 3.13. For n ≥ 0, we have : ( ) ∑n ∑n (α) n (α) F (x; λ; µ; ν) = F S1(l, k) T (x). (3.2.2) n l n−l k k=0 l=k Further in view of relation (1.23), we deduce the following result from (3.2.2)

Corollary 3.14. For n ≥ 0, we have : ( ) ∑n ∑n (α) n (α) B (x; λ) = B S1(l, k) T (x). (3.2.3) n l n−l k k=0 l=k which for λ = α = 1 gives the result of [12; p.42(Theo. 3.1)].

Also in view of relation (1.24), we deduce the following result from (3.2.2)

Corollary 3.15. For n ≥ 0, we have : ( ) ∑n ∑n (α) n (α) E (x; λ) = E S1(l, k) T (x). (3.2.4) n l n−l k k=0 l=k which for λ = α = 1 gives the result of [12; p.43(Theo. 3.3)].

Furthermore, in view of relation (1.25), we deduce the following result from (3.2.2) 48 Ghazala Yasmin

Corollary 3.16. For n ≥ 0, we have : ( ) ∑n ∑n (α) n (α) G (x; λ) = G S1(l, k) T (x). (3.2.5) n l n−l k k=0 l=k Next, let us assume that

∑n F (α) ; ; 3 2 6 Tk(x) = Cn,k n (x λ µ, ν). ( . . ) k=0 From (1.9) and (3.2.6), we have

( t ) 1 λee −1 + 1 α C = ⟨ (et − 1)k|xn⟩ n,k k! 2µ (et − 1)ν ( t ) n ( ) 1 e −1 1 α ∑ λe + n n−l = ⟨ S2(l, k)|x ⟩ 2µα (et − 1)ν l l=k ( ) n∑−k et−1 1 α ∑m j n (λe + ) να t m = S2(n − m, k)⟨ | B x ⟩ m tνα j j! m=0 j=0 ( )( ) ( ) n∑−k ∑m ∑α n m να α α−p et−1 p j = S2(n − m, k)B ⟨ (−λ) (e ) |x ⟩ m j m−j p m=0 j=0 p=0 ( )( )( ) n∑−k ∑m ∑α ∑∞ 1 n m α α−p να (p) = (−λ) S2(n − m, k) B Bell . m j p m−j j+να (j + να)! m=0 j=0 p=0 l=0 (3.2.7) Therefore, by (3.2.6) and (3.2.7), we obtain the following theorem

Theorem 3.17. For n ≥ 0, we have : ( )( )( ) ∑n n∑−k ∑m ∑α α−p n m α (−λ) S2(n − m, k) T (x) = Bνα Bell(p) F (α)(x; λ; µ, ν). k m j p (j + να)! m−j j+να n k=0 m=0 j=0 p=0 (3.2.8) Further in view of relation (1.23), we deduce the following result from (3.2.8)

Corollary 3.18. For n ≥ 0, we have : ( )( )( ) ∑n n∑−k ∑m ∑α α−p n m α (λ) S2(n − m, k) T (x) = Bα Bell(p) B(α)(x; λ). k m j p (j + α)! m−j j+α n k=0 m=0 j=0 p=0 (3.2.9) which for λ = α = 1 gives the result of [12; p.43(Theo. 3.2)].

Also in view of relation (1.24), we deduce the following result from (3.2.8)

Corollary 3.19. For n ≥ 0, we have : ( )( )( ) ∑n n∑−k ∑m ∑α α−p n m α (−λ) S2(n − m, k) T (x) = Bell(p)E(α)(x; λ). (3.2.10) k m j p j! j n k=0 m=0 j=0 p=0 Furthermore, in view of relation (1.25), we deduce the following result from (3.2.8) SOME IDENTITIES OF THE APOSTOL TYPE POLYNOMIALS ... 49

Corollary 3.20. For n ≥ 0, we have : ( )( )( ) ∑n n∑−k ∑m ∑α α−p n m α (−λ) S2(n − m, k) T (x) = Bα Bell(p) G(α)(x; λ). k m j p (j + α)! m−j j+α n k=0 m=0 j=0 p=0 (3.2.11) In the next section, certain results related to some mixed form of Apostol type polynomials are explored.

4 Concluding Remarks

As a remark, in this section we consider recently introduced mixed form of Apostol type poly- nomials dened as Hermite Apostol type polynomials (HATP) and investigate the properties of these polynomials which are derived from umbral calculus. We can establish connection between our polynomials and several known families of polynomials. For example we explore the rela- tion involving Hermite Apostol type polynomials with Apostol type polynomials and Bernoulli polynomials.

(α) The HATP H Fn (x; λ; µ; ν) are dened by the generating function [13]

( µ ν ) ∑∞ n 2 t α 2 t e2xt−t = F (α)(x; λ; µ, ν) . (4.1) λet + 1 H n n! n=0

(α) (α) where H Fn (λ; µ, ν) = H Fn (0; λ; µ, ν) denotes the Hermite Apostol type numbers dened by.

( µ ν ) ∑∞ n 2 t α 2 t e−t = F (α)(λ; µ, ν) . (4.2) λet + 1 H n n! n=0 From (4.1) and (1.6), we note that ( ( ) ) 2 z α t λe + 1 t F (α)(x; λ; µ, ν) ∼ e 4 , . (4.3) H n 2µ tν 2 Also, by (1.2), (4.1)and (4.3), we have ( ) 2 µ ν α − t 2 t F (α)(x; λ; µ, ν) = e 4 (2x)n H n λez + 1

2 − t e 4 = F (α)(x; λ; µ, ν) 2n n

[ n ] ∑4 ( )2m 1 −1 (α) = (n)2 F (x; λ; µ, ν) m!2n 2 m n−2m m=0

[ n ] ( ) ∑4 n (−1)m (2m)! F (α)(x; λ; µ, ν) = F (α) (x; λ; µ, ν). (4.4) H n 2m m! 2n−2m n−2m m=0 Further in view of relation (1.23), we deduce the following result from (4.4)

Corollary 4.1. For n ≥ 0, we have :

[ n ] ( ) ∑4 n (−1)m (2m)! B(α)(x; λ) = B(α) (x; λ). (4.5) H n 2m m! 2n−2m n−2m m=0 50 Ghazala Yasmin

Also in view of relation (1.24), we deduce the following result from (4.4)

Corollary 4.2. For n ≥ 0, we have :

[ n ] ( ) ∑4 n (−1)m (2m)! E(α)(x; λ) = E(α) (x; λ). (4.6) H n 2m m! 2n−2m n−2m m=0 Furthermore, in view of relation (1.25), we deduce the following result from (4.4)

Corollary 4.3. For n ≥ 0, we have :

[ n ] ( ) ∑4 n (−1)m (2m)! G(α)(x; λ) = G(α) (x; λ). (4.7) H n 2m m! 2n−2m n−2m m=0 Let us consider the following two Sheffer sequences: ( ( ) ) 2 z α t λe + 1 t F (α)(x; λ; µ, ν) ∼ e 4 , . n 2µ tν 2 and (( ) ) λez − 1 r B(r)(x) ∼ , z . (4.8) n z Let us assume that ∑n F (α) ; ; B(r) 4 9 H n (x λ µ, ν) = Cn,k n (x). ( . ) k=0 Then, by (1.8) and (1.9), we get ( ) ( ) z r 2 µ ν α 1 e − 1 − z 2 z C = ⟨ zm|e 4 (x)n⟩ n,k m! z λez + 1 ( ) n∑−m n−m n ( l ) (α) = S2(l + r, r) F . (4.10) m l+r H n−m−l l=0 r Therefore, by (4.9) and (4.10), we obtain the following theorem.

Theorem 4.4. For n ≥ 1 and α ∈ Z+, we have : ( ) ∑n n∑−m n−m (α) n ( l ) (α) (r) F (x; λ; µ, ν) = S2(l + r, r) F B (x). (4.11) H n m l+r H n−m−l n k=0 l=0 r Further, we can derive other identities involving HATP and other families of polynomials. In our next investigation we established certain properties of other polynomials using umbral techniques.

References

[1] J. Agapito Ruiz, Riordan arrays from an umbral symbolic viewpoint, Bol. Soc. Port. Mater. 58 (2012)(spe- cial issue). [2] T.M. Apostol, On the Lerch Zeta function, Pacic J. Math. 1 (1951) 161-167. [3] S. Araci, Novel identities involving Genocchi numbers and polynomials arising from applications of um- bral calculus, Appl. Math. Comput. 233 (2014) 599-607. [4] A. Bayad, Fourier expansions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Math. Comp. 80 (2011) 2219-2221. SOME IDENTITIES OF THE APOSTOL TYPE POLYNOMIALS ... 51

[5] L.C. Biedenharn, R.A. Gustafson, M.A. Lohe, J.D. Louck, S.C. Milne, Special functions and group theory in theoretical physics, in: Special Functions: Group Theoretical Aspects and Applications, 1984, 129– 162. [6] L.C. Biedenharn, R.A. Gustafson, S.C. Milne, An umbral calculus for polynomials characterizing u(n) tensor products, Adv. Math. 51 (1984) 36–90. [7] A. Di Bucchianico, D. Loeb, A selected survey of umbral calculus, Electron. J. Combin. 2 (2000) DS3. [8] S. Chen, Y. Cai, Q.M. Luo, An extension of generalized Apostol-Euler polynomials, Adv. Difference Equ. (2013) 61. 10 pp. [9] G. Dattoli, D. Levi, p. Winternitz, Heisenberg algebra, umbral calculus and orthogonal polynomials, J. Math. Phys. 49 (5) (2008):053509. [10] R. Dere, Y. Simsek, Application of umbral algebra to some special polynomials, Adv. Stud. Contemp. Math. 22(3) (2012) 433-438. [11] A. Erdelyi,´ W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, Vol. III, McGraw-Hill Book Company, New York, Toronto and London, 1955. [12] O. Herscovici and T. Mansour, Identities involving Touchard polynomials derived from umbral calculus, Adv. Stud. Contem. Math. 25 (1) (2015) 39-46. [13] S. Khan, G. Yasmin, M. Riyasat, Certain results for the 2-variable Apostol type and related polynomials, Comput. Math. Appl. 69 (2015) 1367-1382. [14] D.S. Kim, T. Kim, Some identities of Frobenius-Euler polynomials arising from umbral calculus, Adv. Differ. Equ. 196 (2012) 2012;196 [15] D.S. Kim, T. Kim, D.V. Dolgy, S.H. Rim, Some new identities of Bernoulli, Euler and Hermite polyno- mials arising from umbral calculus, Adv. Differ. Equ. 73 (2013) 2013;73 [16] D.S. Kim, T. Kim, D.V. Dolgy, S.H. Rim, Some identities Higher-order Bernoulli, Euler and Hermite polynomials arising from umbral calculus, J. Inequal. Appl. (2013) 211. 10 pp. [17] V. Kurt, Some symmetry identities for the unied Apostol type polynomials and multiple power sums, Filomat 30(3) (2016) 583-592. [18] D.Q. Lu, Q.M. Luo, Some properties of the generalized Apostol type polynomials, Boundary Value Prob- lems (2013) 2013;64. [19] D.Q. Lu, H.M. Srivastava, Some series identities involving the generalized Apostol type and related poly- nomials, Comput. Math. Appl. 62 (2011) 3591-3602. [20] D.Q. Lu, Chang-He Xiang, Q.M. Luo, Some results for Apostol-type polynomials associated with umbral algebra, Adv. Difference Equ. (2013) 201. 13 pp. [21] Q.M. Luo, Apostol-Euler polynomials of higher order and the Gaussian hypergeometric function, Tai- wanese J. Math. 10(4) (2006) 917-925. [22] Q.M. Luo, The multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order, Integ. Trans. Spec. Funct. 20 (2009) 377-391. [23] Q.M. Luo, Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials, Math. Comp. 78(2009) 2193-2208. [24] Q.M. Luo, Extension for the Genocchi polynomials and its Fourier expansions and integral representa- tions, Osaka J. Math. 48 (2011) 291-309. [25] Q.M. Luo, The multiplication formulas for the Apostol-Genocchi polynomials, Utilitas Mathematica 89 (2012) 179-191. [26] Q.M. Luo, The multiplication formulas for the Apostol type polynomials and the multiple alternating sum, Math. Notes 91 (2012) 54-65. [27] Q.M. Luo, H.M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomi- als, J. Math. Anal. Appl. 308(1) (2005) 290-302. [28] Q.M. Luo, H.M. Srivastava, Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput. 217 (2011) 5702-5728. [29] T. Mansour and M. Schork, The generalized Touchard polynomials revisted, Appl. Math. Computat. 219 (2013) 9978-9991. [30] Luis M. Navas, Francisco J. Ruis, Juan L. Varona, Asymtotic estimates for Apostol-Bernoulli and Apostol- Euler polynomials, Math. Comp. 81 (2011) 1707-1722. [31] T.J. Robinson, Formal calculus and umbral calculus, Electron. J. Combin. 17 (1)(2010) 31 pp. [32] S. Roman, G.C. Rota, The umbral calculus, Adv. Math. 27(2) (1978) 95-188. [33] S. Roman, The Umbral Calculus, Academic Press, New York, 1984. [34] J. Sandor, B. Crstici, Handbook of Number Theory, Kluwer Academic, Dordrecht 2004. 52 Ghazala Yasmin

[35] W. Shao, Y. He, Some formulas for the generalized Apostol-type polynomials and numbers, J. Nonlinear Sci. Appl. 9(2016) 2511-2519. [36] H.M. Srivastava Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Camb. Philos. Soc. 129 (2000) 77-84.

Author information Ghazala Yasmin, Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh, India. E-mail: [email protected]

Received: October 30, 2016.

Accepted: December 19, 2016.