Generating functions for Jacobi and related polynomials
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- Proc. Amer. Math. Soc. 32 (1972), 179-186 Request permission
Abstract:
In the present paper, we have established the following relation involving Kampe de Feriet’s double hypergeometric function of superior order \[ F\left [ {\begin {array}{*{20}{c}} p \\ 1 \\ q \\ 0 \\ \end {array} \left | {\begin {array}{*{20}{c}} {{a_1}, \cdots ,{a_p}} \\ {b, - b’} \\ {{c_1}, \cdots ,{c_q}} \\ \cdots \\ \end {array} } \right | - xy, - y} \right ] = \sum \limits _{n = 0}^\infty {\frac {{\prod \limits _{j = 1}^p {{{({a_j})}_n}} }}{{\prod \limits _{j = 1}^q {{{({b_j})}_n}} }}\frac {{{{( - 1)}^n}{{( - b’)}_n}}}{{n!}}{\;_2}{F_1}\left [ {\begin {array}{*{20}{c}} { - n,b;} \\ {1 + b’ - n;} \\ \end {array} x} \right ]{y^n},} \] which yields a number of interesting generating formulae for Jacobi and related polynomials. A large number of special cases have been also discussed.References
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Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 32 (1972), 179-186
- MSC: Primary 33A65
- DOI: https://doi.org/10.1090/S0002-9939-1972-0348161-X
- MathSciNet review: 0348161