On power series of products of special functions (Q540294)

The author gives some methods of derivation of power expansion for products and powers of various elementary and special functions. The principal formula that the author used to obtain power series expansions of products of special function is \[ \begin{multlined} {}_p F_q \binom {(a_p); z} {(b_q)} {}_r F_s \binom{(c_r); az} {(d_s)} \\ = \sum_{k=0}^\infty {{\prod (a_p)_k} \over {\prod (b_q)_k}} {}_{q+r+1}F_{p+s} \binom{-k, 1-k-(b_q), (c_r)} {1-k-(a_p), (d_s); (-1)^{p+q+1} a} {{z^k}\over{k!}}, \end{multlined} \] where \({}_{p}F_{q} \binom{(a_p); z} {(b_q)} \) is the generalized hypergeometric function, \((a_p) = a_1, \dots, a_p. \) Also in case that the lower parameters of a hypergeometric function are negative integers or zero, the author uses the so called regularized hypergeometric function defined by \[ { }_{p}\widetilde{F}_{q} \binom{ a_1, a_2, \dots, a_p; z} {b_1, b_2, \dots, b_q } = \sum_{k=0}^\infty {{(a_1)_k (a_2)_k \cdots (a_p)_k}\over {\Gamma(b_1 +k) \Gamma(b_2+k) \cdots \Gamma(b_q +k)}} {{z^k}\over{k!}} \] to find special cases. Some examples demonstrated in this paper are power series of products of \(e^z \) and \(H_{2n} (az), \) where \(H_n(z) \) is the Hermite polynomial, products of \(\arcsin z \) and \(T_{2n} (az), \) where \(T_{2n} (z) \) is the Chebyshev polynomial, and products of \(P_{2m} (z) \) and \(C_{2n+1} ^\lambda (az), \) where \(P_n(z) \) and \(C_n^\lambda (z) \) are the Legendre and Gegenbauer polynomials, respectively.