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A New Class of Hermite-Fubini Polynomials and Its Properties

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A New Class of Hermite-Fubini Polynomials and Its Properties Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 30 April 2019 doi:10.20944/preprints201904.0333.v1 A new class of Hermite-Fubini polynomials and its properties Waseem A. Khan1 and Nisar K S2 1Department of Mathematics, Faculty of Science, Integral University, Lucknow-226026, (India) 2Department of Mathematics, College of Arts and Science-Wadi Al dawaser, Prince Sattam bin Abdulaziz University, Riyadh region 11991, Saudi Arabia E-mail: waseem08 khan@rediffmail.com, [email protected] Abstract. In this paper, we introduce a new class of Hermite-Fubini numbers and polynomials and investigate some properties of these polynomials. We establish sum- mation formulas of these polynomials by summation techniques series. Furthermore, we derive symmetric identities of Hermite-Fubini numbers and polynomials by using generating functions. Keywords: Hermite polynomials, Fubini numbers and polynomials, Hermite-Fubini polynomials, summation formulae, symmetric identities. 2010 Mathematics Subject Classification.: 11B68, 11B75, 11B83, 33C45, 33C99. 1. Introduction As is well known, the 2-variable Hermite Kamp´ede F´erietpolynomials (2VHKdFP) Hn(x; y) [1, 3] are defined as [ n ] X2 yrxn−2r H (x; y) = n! : (1:1) n r!(n − 2r)! r=0 It is clear that 1 H (2x; −1) = H (x; H (x; − ) = He (x);H (x; 0) = xn; n n n 2 n n where Hn(x) and Hen(x) being ordinary Hermite polynomials. The Hermite polynomial Hn(x,y) (see ([9, 10]) is defined by means of the fol- lowing generating function as follows: X1 n 2 t ext+yt = H (x; y) : (1:2) n n! n=0 Geometric polynomials (also known as Fubini polynomials) are defined as follows (see [2]): { } Xn n F (x) = k!xk; (1:3) n k { } k=0 n where is the Stirling number of the second kind (see [5]). k For x = 1 in (1.3), we get nth Fubini number (ordered Bell number or geometric number) Fn [2, 4, 5, 6, 8, 12] is defined by { } Xn n F (1) = F = k!: (1:4) n n k k=0 © 2019 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 30 April 2019 doi:10.20944/preprints201904.0333.v1 2 The exponential generating functions of geometric polynomials is given by (see [2]): 1 1 X tn = F (x) ; (1:5) 1 − x(et − 1) n n! n=0 and related to the geometric series (see [2]): ( ) 1 d m 1 X 1 x x = kmxk = F ( ); j x j< 1: dx 1 − x 1 − x m 1 − x k=0 Let us give a short list of these polynomials and numbers as follows: 2 2 3 2 3 4 F0(x) = 1;F1(x) = x; F2(x) = x+2x ;F3(x) = x+6x +6x ;F4(x) = x+14x +36x +24x ; and F0 = 1;F1 = 1;F2 = 3;F3 = 13;F4 = 75: Geometric and exponential polynomials are connected by the relation (see [2]): Z 1 −λ Fn(x) = ϕn(x)e dλ. (1:6) 0 Recently, Pathan and Khan [9] introduced two variable Hermite-Bernoulli poly- nomials is defined by means of the following generating function: ( ) α X1 n t 2 t ext+yt = B(α)(x; y) : (1:7) et − 1 H n n! n=0 On setting α = 1 in (1.7), the result reduces to known result of Dattoli et al. [3]. The manuscript of this paper as follows: In section 2, we consider generat- ing functions for Hermite-Fubini numbers and polynomials and give some properties of these numbers and polynomials. In section 3, we derive summation formulas of Hermite-Fubini numbers and polynomials. In Section 4, we construct a symmetric identities of Hermite-Fubini numbers and polynomials by using generating functions. 2. A new class of Hermite-Fubini numbers and polynomials In this section, we define three-variable Hermite-Fubini polynomials and obtain some basic properties which gives us new formula for H Fn(x; y; z). Moreover, we shall consider the sum of products of two Hermite-Fubini polynomials. The sum of products of various polynomials and numbers with or without binomial coefficients have been studied by (see [2, 4, 5, 6, 8]): We introduce 3-variable Hermite-Fubini polynomials by means of the following generating function: 2 1 ext+yt X tn = F (x; y; z) : (2:1) 1 − z(et − 1) H n n! n=0 It is easily seen from definition (2.1), we have H Fn(0; 0; z) = Fn(z); H Fn(0; 0; 1) = Fn: For y = 0 in (2.1), we obtain 2-variable Fubini polynomials which is defined by Kargin [8]. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 30 April 2019 doi:10.20944/preprints201904.0333.v1 3 1 ext X tn = F (x; z) : (2:2) 1 − z(et − 1) n n! n=0 When investigating the connection between Hermite polynomials Hn(x; y) and Fubini polynomials Fn(z) of importance is the following theorem. Theorem 2.1. The following summation formula for Hermite-Fubini polynomials holds true: 2 1 e−yt X (−1)nt2n [cos xt(z + 1) − z cos(t − xt)] = F (x; y; z) (2:3) Ω H 2n (2n)! n=0 2 1 e−yt X (−1)nt2n+1 [sin xt(z + 1) + z sin(t − xt)] = F (x; y; z) ; (2:4) Ω H 2n+1 (2n + 1)! n=0 where Ω = [1 − z(cos t − 1)]2 + [z sin t]2. Proof. On separating the power series on r.h.s. of (2.1) in to their even and odd terms by using the elementary identity X1 X1 X1 f(n) = f(2n) + f(2n + 1) n=0 n=0 n=0 and then replacing t by it where i2 = −1 and equating the real and imaginary parts in the resulting equation, we get the summation formulae (2.2) and (2.3). Remark 2.1. On setting x = y = 0, z = 1 in (2.3) and (2.4), we get the following well-known classical results involving Fubini numebrs. Corollary 2.1. The following summation formula for Hermite-Fubini polynomials holds true: 1 2 − cos t X (−1)nt2n = F (2:5) 5 − 4 cos t 2n (2n)! n=0 1 sin t X (−1)nt2n+1 = F : (2:6) 5 − 4 cos t 2n+1 (2n + 1)! n=0 Theorem 2.2. For n ≥ 0, the following formula for Hermite-Fubini polynomials holds true: ( ) Xn n F (x; y; z) = F − (z)H (x; y): (2:7) H n r n m m m=0 Proof. Using definition (2.1), we have 1 2 X tn ext+yt F (x; y; z) = H n n! 1 − z(et − 1) n=0 1 1 X tn X tm = F (z) H (x; y) n n! m m! n=0 m=0 ( ) ! X1 Xn n tn = F − (z)H (x; y) : r n m m n! n=0 m=0 tn Comparing the coefficients of n! yields (2.7). Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 30 April 2019 doi:10.20944/preprints201904.0333.v1 4 Theorem 2.2. For n ≥ 0, the following formula for Hermite-Fubini polynomials holds true: Hn(x; y) = H Fn(x; y; z) − zH Fn(x + 1; y; z) + zH Fn(x; y; z): (2:8) Proof. We begin with the definition (2.1) and write t 2 1 − z(e − 1) 2 ext+yt = ext+yt 1 − z(et − 1) xt+yt2 t e z(e − 1) 2 = − ext+yt 1 − z(et − 1) 1 − z(et − 1) Then using the definition of Kamp´ede F´erietgeneralization of the Hermite polyno- mials Hn(x; y) and (2.1), we have 1 1 X tn X tn H (x; y) = [ F (x; y; z) − z F (x + 1; y; z) + z F (x; y; z)] : n n! H n H n H n n! n=0 n=0 tn Finally, comparing the coefficients of n! , we get (2.8). Theorem 2.3. For n ≥ 0 and z1 =6 z2, the following formula for Hermite-Fubini polynomials holds true: ( ) Xn n F − (x ; y ; z ) F (x ; y ; z ) k H n k 1 1 1 H k 2 2 2 k=0 z F (x + x ; y + y ; z ) − z F (x + x ; y + y ; z ) = 2H n 1 2 1 2 2 1H n 1 2 1 2 1 : (2:9) z2 − z1 Proof. The products of (2.1) can be written as X1 X1 2 2 tn tk ex1t+y1t ex2t+y2t H Fn(x1; y1; z1) H Fk(x2; y2; z2) = t t n! k! 1 − z1(e − 1) 1 − z2(e − 1) n=0 k=0 ( ) ! X1 Xn n tn F − (x ; y ; z ) F (x ; y ; z ) k H n k 1 1 1 H k 2 2 2 n! n=0 k=0 2 2 z e(x1+x2)t+(y1+y2)t z e(x1+x2)t+(y1+y2)t = 2 − 1 z − z 1 − z (et − 1) z − z 1 − z (et − 1) ( 2 1 1 2 1 2 ) z F (x + x ; y + y ; z ) − z F (x + x ; y + y ; z ) tn = 2H n 1 2 1 2 2 1H n 1 2 1 2 1 : z2 − z1 n! tn By equating the coefficients of n! on both sides, we get (2.9). Theorem 2.4. For n ≥ 0, the following formula for Hermite-Fubini polynomials holds true: zH Fn(x + 1; y; z) = (1 + z)H Fn(x; y; z) − Hn(x; y): (2:10) Proof.
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