Positivity of the Poisson Kernel for the Continuous q -Jacobi Polynomials and Some Quadratic Transformation Formulas for Basic Hypergeometric Series
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 re...
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| Published in: | SIAM journal on mathematical analysis Vol. 17; no. 4; pp. 970 - 999 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Philadelphia, PA
Society for Industrial and Applied Mathematics
01.07.1986
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| Subjects: | |
| ISSN: | 0036-1410, 1095-7154 |
| Online Access: | Get full text |
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| Summary: | 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. |
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| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 content type line 14 |
| ISSN: | 0036-1410 1095-7154 |
| DOI: | 10.1137/0517069 |