Remarks on Homogeneous Al-Salam and Carlitz Polynomials

Abstract

Several multilinear generating functions of the homogeneous Al-Salam and Carlitz polynomials are derived fromq-operator. In addition, two interesting relationships of product of this kind of polynomials are obtained.

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

“…For c = d = e = f = g = 0, and b→ 1 b , y → yb, b → 0, then Equation (16) reduces to the concluding remarks in [21]. For a = b = c = d = e = f = g = 0 in Theorem 1, Equations ( 16) and (20) reduce to Equations (11) and (12), respectively.…”

“…For more information on q-exponential operators, see the details in [7,8,[10][11][12][13][14][15][16].…”

“…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.…”

“…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].…”