Details

Autor(en) / Beteiligte

• GASPER, G
• RAHMAN, M

Titel

Positivity of the Poisson kernel for the continuous q-Jacobi polynomials and some quadratic transformation formulas for basic hypergeometric series

Ist Teil von

• SIAM journal on mathematical analysis, 1986-07, Vol.17 (4), p.970-999

Ort / Verlag

Philadelphia, PA: Society for Industrial and Applied Mathematics

Erscheinungsjahr

1986

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• A nonterminating extension of the Sears-Carlitz quadratic transformation formula for a well-posed _3 \phi _2 $ series with an arbitrary argument is obtained as a sum of two balanced _5 \phi _4 $ series. This is then extended to a very well-poised _5 \phi _4 $ series with arbitrary argument. These results are used to derive some generating functions for the $q$-Wilson polynomials $p_n (x;a,b,c,d;q)$ when $ad = bc$ and an expression for the Poisson kernel $K_t (x,y;a,b,c,{{bc} / {a;}}q)$ as a sum of three sums of very well-poised _10 \phi _9 $ series which clearly demonstrates its positivity for $0 \leqq t < 1$, $0 \leqq q < 1$ in the continuous $q$-Jacobi case when $a = q^{{\alpha / 2} + {1 / 4}} $, $b = q^{{\alpha / 2} + {3 / 4}}$, $c = q^{{\beta / 2} + {1 / 4}} $ and $\alpha $, $\beta > - 1$. Additional quadratic transformation formulas are derived, along with $q$-analogues of Watson's and Whipple's summation formulas.