The role of the M-Wright function in bi-orthogonality of the fractional Bernoulli and Euler polynomials

In this article, we derive the Sheffer polynomials {S_m(x, y)}_m=1^∞ in two variables as the coefficient set of the generating function A(t, y)e^xt, where A(s, y) is a complex function with respect to complex variable s and y ϵ R. When the function A(s, y) is entire, using the inverse Mellin transform we get the coefficient set, and when the function A(s, y) has a branch point at zero point s = 0, using the M-Wright function, we derive the coefficient set. Moreover, as special cases of this set for the Bernoulli and Euler polynomials, bi-orthogonality of these polynomials with their associated functions is discussed. The bi-orthogonality relations are given in terms of the product of the M-Wright function and the Sheffer polynomials.