ANALYSIS OF BELL BASED EULER POLYNOMIAL AND THEIR APPLICATION NABIULLAH KHAN AND SADDAM HUSAIN Abstract. In the present article, we study Bell based Euler polynomial of order α and investigate some useful correlation formula, summation formula and derivative formula . Also, we introduce some relation of string number of the second kind. Moreover, we drive several important formulae of bell based Euler polynomial by using umbral calculus. 1. Introduction The polynomials and number play an important role in the multifarious area of science such as mathematics, applied science, physics and engineering sciences and some related research area involving number theory, quantum mechanics, dif- ferential equation and mathematical physics (see [1, 3, 4]). Some of the most important polynomials are Bell polynomial, Genocchi polynomial, Euler polyno- mial, Bernaulii polynomial and Hermite polynomial. We know that, properties of polynomials with the help of umbral calculus studied by many authors. Kim et al. [11] study some properties of Euler polynomials by using umbral Calculus and give some useful identities of Euler polynomials. Kim et al. [12] study Bernoulli, Euler and Abel polynomials by using umbral calculus and gives interesting identities of it. Kim et al. [14] investigate useful identities for partially degenerate Bell Number and polynomials associated with umbral calculus and Kim et al. [15] drived some important identities and properties of umbral calculus associated with degenerate orderd Bell number and polynomials. arXiv:2104.09129v1 [math.NT] 19 Apr 2021 Recently, S. Araci et al. [7] defined a Bell based Bernaoulli polynomials. Moti- vated by above mention work in the present article we introduce Bell based Euler polynomials and investigate some useful correlation formula, summation formula and derivative formula of Bell based Euler polynomial of order α. Also, we acquire some implicit summation formula and some special cases of Bell based Euler poly- nomials. Further, we drived some interesting result of Bell based Euler polynomials associate with umbral calculus. 2010 Mathematics Subject Classification. 11B68, 33B15, 33C05, 33C10, 33C15, 33C45, 33E20. Key words and phrases. Bell polynomial, Euler polynomial, String polynomial, String number of second kind, Sheffer sequence. 1 2 N. U. KHAN AND S. HUSAIN 2. Preliminaries In the present paper, we take the symbols Z, N, N0, R, and C to be set of integer, set of natural number, set of non negative integers, set of real number and set of complex number respectively. The bivariate Bell polynomials are describe by the following generating function, which is defined as follows: n t t B (x; y) = ext ey(e −1). (2.1) n n! n≥0 X When we take x = 0, Bn(0; y)= Bn(y) are called classical Bell polynomials (or exponential polynomials) which is describe by following generating function (see [2, 3, 4, 13, 17]) which is defined as follows: n t t B (y) = ey(e −1). (2.2) n n! n≥0 X If we take y = 1 in (2.2) i.e. Bn(0;1) = Bn(1) = Bn are called Bell number which is defined as follows (see [2, 3, 4, 13, 17]): n t t B = e(e −1). (2.3) n n! n≥0 X The generating function of an Euler polynomials of order α (see[8, 9, 10, 11, 18, 19]) is as folows: tn 2 α E (α)(x) = ext (|t| < 2π). (2.4) n n! et +1 n≥0 X (α) (α) If we take x = 0 in (2.4) i.e. En (0) = En are called Euler number (see[8, 9, 10, 11, 18, 19]) which is shown as follows: tn 2 α E (α) = . (2.5) n n! et +1 n≥0 X The generating function of second kind string polynomials S2(n, k; x) and string number S2(n, k) are defined as (see[2, 3]): tn et − 1 k S (n, k; x) = etx. (2.6) 2 n! k! n≥0 X When x = 0 in (2.6) i.e. S2(n, k;0) = S2(n, k) are called string number and defined by following exponential generating function (see[2, 3]): tn et − 1 k S (n, k) = . (2.7) 2 n! k! n≥0 X ANALYSIS OF BELL BASED EULER POLYNOMIAL AND THEIR APPLICATION 3 3. Bell based Euler polynomials and Number In this segment, we introduced Bell based Euler polynomials of order α and investigate numerous correlation formulae like implicit summation formulae, de- rivative formulae. For any n ∈ N and α ∈ C, we define Bell based Euler polynomial of order α as: n α (α) t 2 xt+y(et−1) BE (x; y) = e (|t| < 2π). (3.1) n n! et +1 n≥0 X If x=0 and y=1 in (3.1) then we get a Bell Based Euler number of order α, which is defined as follows: n α (α) t 2 (et−1) BE = e (|t| < 2π). (3.2) n n! et +1 n≥0 X 3.1. Special Cases: In this section, we introduce some special types of Bell based Euler polynomials of order α, which is obtain by putting particular value in (3.1) and defined as follows: (1) If we choose x=0 in (3.1), we get Bell based Euler polynomials of order α, which are an extension of Euler polynomials of order α defined in (2.4) as follows: n α (α) t 2 y(et−1) BE (y) = e . n n! et +1 n≥0 X (2) In case y=0 in (3.1) the Bell based Euler polynomials of order α reduced (α) to the famililar Euler polynomials En (x) of order α defined in (2.4) n α (α) t 2 xt BE (x) = e . n n! et +1 n≥0 X (α) (3) In case y=0 and α=1 in (3.1) the Bell Based Euler polynomials BEn (x; y) reduced to usual Euler polynomials En(x) defined as: n t 2 xt BE (x) = e . n n! et +1 n≥0 X Theorem 3.1. The following relation hold true for α ∈ C and n ∈ N; n n E (α)(x; y)= E (α)(x)B (y). (3.3) B n k k n−k k X=0 Proof. By using relation (3.1), we have 4 N. U. KHAN AND S. HUSAIN n α (α) t 2 xt+y(et−1) BE (x; y) = e n n! et +1 n≥0 X α 2 t = ext ey(e −1) et +1 n o tk tn = E (α)(x) B (y) . k k! n n! (k≥ ) (n≥0 ) X0 X Now, using series rearrangement method, we get n n n (α) t n (α) t BE (x; y) = E (x)B − (y) . n n! k k n k n! n≥0 n≥0 (k ) X X X=0 By equating same power of t both side, we get desired result (3.3). Theorem 3.2. Bell based Euler polynomials of order α satisfy following relation for any α ∈ C and n ∈ N: n n E (α)(x; y)= E (α)B (x; y). (3.4) B n k k n−k k X=0 Proof. By using generating function (3.1), we have n α (α) t 2 xt+y(et−1) BE (x; y) = e n n! et +1 n≥0 X α 2 t = ext+y(e −1) et +1 n o tk tn = E (α) B (x; y) k k! n n! (k≥ ) (n≥0 ) X0 X After applying series rearrangement technique and compare same power of t, we get desired result (3.4). Theorem 3.3. If α ∈ C and n ∈ N, following relation hold true; n n E (α)(x; y)= E (α)(y) xn−k. (3.5) B n k k k X=0 ANALYSIS OF BELL BASED EULER POLYNOMIAL AND THEIR APPLICATION 5 Proof. Using relation (3.1), we have n α (α) t 2 xt+y(et−1) BE (x; y) = e n n! et +1 n≥0 X α 2 t = ey(e −1) ext et +1 k n (α) t (xt) = BE (y) k k! n! (k≥ ) (n≥0 ) X0 X n n+k (α) x t = BE (y) . k n! k! (n≥0 k≥ ) X X0 By using series rearrangement, we get n n n (α) t n (α) n−k t BE (x; y) = BE (y) x . n n! k k n! n≥0 n≥0 (k ) X X X=0 By equating same power of t both side, we get desired result (3.5). 4. Implicit summation formulae In this section, we discus some useful implicit summation formulae for Bell Based Euler polynomials of order α, which is defined in following theorem as follows: Theorem 4.1. If α1,α2 ∈ C and n ∈ N, following relation hold true; n n E (α1+α2)(x + x ; y + y )= E (α1)(x ; y ) E (α2)(x ; y ). (4.1) B n 1 2 1 2 k B k 1 1 B n−k 2 2 k X=0 Proof. Using the following identity α1+α2 2 t e(x1+x2)t+(y1+y2)(e −1) et +1 2 α1 2 α2 = ex1t + y (et − 1) ex2t + y (et − 1) . et +1 1 et +1 2 By using generating function (3.1), we have n (α1+α2) t BE (x + x ; y + y ) n 1 2 1 y n! n≥0 X 6 N. U. KHAN AND S. HUSAIN α1+α2 2 t = e(x1+x2)t+(y1+y2)(e −1) et +1 2 α1 2 α2 = ex1t + y (et − 1) ex2t + y (et − 1) et +1 1 et +1 2 k n (α1) t (α2) t = BE (x ; y ) BE (x ; y ) k 1 1 k! n 2 2 n! (k≥ ) (n≥0 ) X0 X n+k (α1) (α2) t = BE (x ; y ) BE (x ; y ) k 1 1 n 2 2 n!k! (n≥0 k≥ ) X X0 Using series rearrangement technique, we obtain n n n (α1+α2) t n (α1) (α2) t BE (x + x ; y + y ) = BE (x ; y ) BE (x ; y ) n 1 2 1 y n! k k 1 1 n−k 2 2 n! n≥0 n≥0 (k ) X X X=0 Now, equating same power of t both side, we get desired result (4.1).