Recurrence relations for hypergeometric functions of unit argument

Lewanowicz, Stanisław

doi:10.1090/S0025-5718-1985-0804941-2

Recurrence relations for hypergeometric functions of unit argument
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Math. Comp. 45 (1985), 521-535 Request permission

Corrigendum: Math. Comp. 48 (1987), 853.
Corrigendum: Math. Comp. 48 (1987), 853-854.

Abstract:

We show that the generalized hypergeometric function \[ {P_n}:{ = _{p + 3}}{F_{p + 2}}\left ( {\left . {\begin {array}{*{20}{c}} { - n,n + \lambda ,{a_p},1} \\ {{b_{p + 2}}} \\ \end {array} } \right |1} \right )\quad (n \geqslant 0)\] satisfies a nonhomogeneous recurrence relation of order $p + \sigma$, where $\sigma = 0$ when $_{p + 3}{F_{p + 2}}(1)$ is balanced, and $\sigma = 1$ otherwise. Also, for \[ {U_n}: = \frac {{{{({c_{q + 1}})}_n}}}{{{{({d_q})}_n}{{(n + \lambda )}_n}}}{ _{q + 2}}{F_{q + 1}}\left ( {\left . {\begin {array}{*{20}{c}} {n + {c_{q + 2}}} \\ {n + {d_q},2n + \lambda + 1} \\ \end {array} } \right |1} \right )\quad (n \geqslant 0)\] a homogeneous recurrence relation of order $q + 1$ is given.

References

Higher Transcendental Functions

Shyam L. Kalla, Salvador Conde, and Yudell L. Luke, Integrals of Jacobi functions, Math. Comp. 38 (1982), no. 157, 207–214. MR 637298, DOI 10.1090/S0025-5718-1982-0637298-8
Samuel Karlin and James L. McGregor, The Hahn polynomials, formulas and an application, Scripta Math. 26 (1961), 33–46. MR 138806

The Special Functions and Their Approximations

Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. MR 0201688
J. A. Wilson, Three-term contiguous relations and some new orthogonal polynomials, Padé and rational approximation (Proc. Internat. Sympos., Univ. South Florida, Tampa, Fla., 1976) Academic Press, New York, 1977, pp. 227–232. MR 0466671
James A. Wilson, Some hypergeometric orthogonal polynomials, SIAM J. Math. Anal. 11 (1980), no. 4, 690–701. MR 579561, DOI 10.1137/0511064
Jet Wimp, Recursion formulae for hypergeometric functions, Math. Comp. 22 (1968), 363–373. MR 226065, DOI 10.1090/S0025-5718-1968-0226065-8
Jet Wimp, The computation of $_{3}F_{2}(1)$, Internat. J. Comput. Math. 10 (1981/82), no. 1, 55–62. MR 644716, DOI 10.1080/00207168108803266
Jet Wimp, Differential-difference properties of hypergeometric polynomials, Math. Comp. 29 (1975), 577–581. MR 440085, DOI 10.1090/S0025-5718-1975-0440085-3

Comput. Math. Appl.

S. Lewanowicz, On the differential-difference properties of the extended Jacobi polynomials, Math. Comp. 44 (1985), no. 170, 435–441. MR 777275, DOI 10.1090/S0025-5718-1985-0777275-2