An operator approach to the Al-Salam–Carlitz polynomials

Abstract

Abstract. We present an operator approach to Rogers-type formulas and Mehler's formulas for the Al-Salam-Carlitz polynomials U n (x, y, a; q). By using the q-exponential operator, we obtain a Rogers-type formula which leads to a linearization formula. With the aid of a bivariate augmentation operator, we get a simple derivation of Mehler's formula due to by Al-Salam and Carlitz, which requires a terminating condition on a 3 φ 2 series. By means of the Cauchy companion augmentation operator, we obtain Mehler's formula in a similar form, but it does not need the terminating condition. We also give several identities on the generating functions for products of the Al-Salam-Carlitz polynomials which are extensions of formulas for Rogers-Szegö polynomials.Keywords: Al-Salam-Carlitz polynomial, the q-exponential operator, the homogeneous qshift operator, the Cauchy companion operator, the Rogers-type formula, Mehler's formula

Remarks on Homogeneous Al-Salam and Carlitz Polynomials

“…The history of these polynomials may go back to Al-Salam and Carlitz in 1965. Since then, these polynomials have been studied by many mathematicians [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Recently, Cao [7] used Carlitz's -operators to study the following homogeneous Al-Salam and Carlitz polynomials:…”

A Review of q-Difference Equations for Al-Salam–Carlitz Polynomials and Applications to U(n + 1) Type Generating Functions and Ramanujan’s Integrals

“…Wang [35] obtained a q-integral representation of the Al-Salam-Carlitz polynomials. Chen, Saad and Sun [36] deduced several properties of the Al-Salam-Carlitz polynomials. Fang [12] deduced several multilinear generating functions of the homogeneous Al-Salam-Carlitz polynomials via q-operators.…”

“…Srivastava and Arjika [37] obtained bilinear generating functions involving the generalized Al-Salam-Carlitz polynomials. For further information about Al-Salam-Carlitz polynomials, see the details in [12,[33][34][35][36][37][38][39][40][41].…”

“…In this paper, we use Liu's calculus method to research and find the q-difference equations of the general Al-Salam-Carlitz polynomials and to generalize the related results of Al-Salam-Carlitz polynomials in the above references [12,[35][36][37].…”

Generalizations of certain Carlitz’s trilinear and Srivastava–Agarwal type generating functions

“…The author of[10,15,16] derived several generating functions for Rogers-Szegö polynomials by q-exponential operators. For more information about Rogers-Szegö and Hahn polynomials, please refer to[7,8,[10][11][12][14][15][16][17][18][19][20][21][22]. Carlitz [7, Eq.…”