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