\(q\)-derivative of basic hypergeometric series with respect to parameters. (English) Zbl 1195.30038

Ghany, Hossam A.

Int. J. Math. Anal., Ruse 3, No. 33-36, 1617-1632 (2009).

Summary: This article discusses the effect of the Difference operator \(D_q\) on the generalized hypergeometric series \(_r \varphi _s(a_1, \dots a_r; b_1, \dots , b_s; q, z)\) with respect to parameters \(a_1, \dots a_r; b_1, \dots , b_s\) and gives some \(q\)-difference equations satisfied by \(_r \varphi _s\), u-exponential function and \(q\)-Appell’s hypergeometric series. Moreover, I will prove that the basic hypergeometric functions \(_r \Phi _s\) are basically completely monotonic with respect to parameters \(a_i, i = 1, 2, \dots, r\) if the parametr \(a_i\) is less than or equal to unity and the functions \(_r \Phi _s\) have positive \(q\)-derivative of all orders.
Finally, the basic hypergeometric functions \(_r \Phi _s\) are totally basically completely monotonic if all parametrs are less than or equal to unity and the functions \(_r \Phi _s\) have positive \(q\)-derivative of all orders.

MSC:

30D15	Special classes of entire functions of one complex variable and growth estimates
33D15	Basic hypergeometric functions in one variable, \({}_r\phi_s\)
42C15	General harmonic expansions, frames

Keywords:

basic hypergeometric series; \(q\)-difference operators; \(q\)-difference equations