SOME IDENTITIES of the APOSTOL TYPE POLYNOMIALS ARISING from UMBRAL CALCULUS Ghazala Yasmin
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Palestine Journal of Mathematics Vol. 7(1)(2018) , 3552 © Palestine Polytechnic University-PPU 2018 SOME IDENTITIES OF THE APOSTOL TYPE POLYNOMIALS ARISING FROM UMBRAL CALCULUS 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) = S1 k tkjα 2 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 hLjp(x)i. The formal power series α f(t) = S1 k tk 2 F; k=0 k! denes a linear functional on P by setting n hf(t)jx i = αn; (n ≥ 0): (1:1) From (1.1), we note that k n ht jx i = n!δn;k; (n; k ≥ 0); (1:2) where δn;k is the Kronecker symbol (see [10, 14, 15, 32, 33]). k Let S1 hLjx i k. Then, by (1.1), we get h j ni h j ni . 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 hg(t)f(t) jSn(x)i = 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 heytjp(x)i = p(y): Let f(t) 2 F and p(x) 2 P. Then we have hf(t)jxki htkjp(x)i f(t) = S1 tk; p(x) = S1 xk (1:3) k=0 k! k=0 k! and by (1.2), we get: p(k)(0) = htkjp(x)i h1jp(k)(x)i = 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 hf(t)(g(t)jp(x)i = h(g(t)jf(t)p(x)i: Let Sn(x) ∼ (g(t); f(t)). Then we see that 1 yf¯(t) 1 Sk(y) k e = S 0 t ; 8 y 2 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 = h l(f(t)) jx i: (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]: ( ) 1 t α X tn ext = B(α)(x) ; jtj < 2π; (1:10) et − 1 n n! n=0 ( ) 1 2 α X tn ext = E(α)(x) ; jtj < π; (1:11) et + 1 n n! n=0 ( ) 1 2t α X tn ext = G(α)(x) ; jtj < 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) 2 N : N [ f0g 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 2 N0, denoted by Bn (x; λ); λ 2 C, which are dened by the generating function ( ) 1 t α X tn ext = B(α)(x; λ) ; jtj < 2π; when λ = 1; jtj < jlogλj; when λ =6 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 2 N0, denoted by En (x; λ); λ 2 C are introduced by Luo [21] and are dened by the generating function ( ) 1 2 α X tn ext = E(α)(x; λ) ; jtj < jlog(−λ)j; (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 2 N0, denoted by (α) Gn (x; λ); λ 2 C, which are dened by the generating function ( ) 1 2t α X tn ext = G(α)(x; λ) ; jtj < jlog(−λ)j; (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 2 N; λ, µ, ν 2 C) of order α, which are dened by means of the generating function ( ) 1 2µ tν α X tn ext = F (α)(x; λ; µ, ν) ; jtj < jlog(−λ)j (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 ( ) 1 2µ tν α X tn = F (α)(λ; µ, ν) ; jtj < jlog(−λ)j: (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 X1 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 X 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 X 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 Stirling number is given by Xn 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 X1 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.