Applications of quadratic and cubic hypergeometric transformations

Per Karlsson, Thomas Ernst

Abstract


The purpose of this paper is to consider five classes of quadratic and cubic hypergeometric transformations in the spirit of Bailey and Whipple. We shall successfully evaluate several hypergeometric functions, of the types \(_{2}\text{F}_{1}(x)\), \(_{3}\text{F}_{2}(x)\), and \(_{4}\text{F}_{3}(x)\), with each function having one or more free parameters, and with the argument $x$ chosen to equal such unusual values as \(x=\pm 1,-8,\frac 14, -\frac 18\), (these four values having been explored initially by Gessel and Stanton). In each case, companion identities and/or inverse transformations are given, which are sometimes proved by a limiting process for a divergent hypergeometric series. Some of the proofs use the Clausen quadratic formula, Euler reflection formula, Legendre duplication, Gauss multiplication formula, Euler transformation, hypergeometric reversion formula and known hypergeometric summation formulas. The proofs in the terminating case are simpler and can lead to mixed summation formulas, which depend on values of a negative integer. Some of the formulas use the Digamma function and a dimension formula is referred to.

Keywords


Quadratic and cubic hypergeometric transformations; Clausen’s quadratic formula; divergent hypergeometric series; L’Hopital’s rule; dimension formula

Full Text:

PDF

References


Andrews, G. E., Askey, R., Roy R., Special functions, in: Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999.

Bailey, W. N., Products of generalized hypergeometric series, Proc. London Math. Soc. (2) 28 (1928), 242–254.

Champion, P. M., Danielson, L. R., Miksell, S. G., Summation of a special hypergeometric series of type 3F2, Ganita 20(1) (1969), 47–48.

Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G., Higher transcendental functions. Vol. I. Based, in part, on notes left by Harry Bateman, McGraw-Hill Book Company, Inc., New York–Toronto–London, 1953.

Ernst, T., A Comprehensive Treatment of q-calculus, Birkhauser, Basel, 2012.

Gessel, I., Stanton, D., Strange evaluations of hypergeometric series, SIAM J. Math. Anal. 13 (1982), 295–308.

Karlsson, P. W., On some hypergeometric transformations, Panam. Math. J. 10(4) (2000), 59–69.

Karlsson, P., Ernst, T., Corollaries and multiple extensions of Gessel and Stanton hypergeometric summation formulas, Acta Comment. Univ. Tartu. Math. 25(1) (2021), 21–31.

Olver, F. (ed.), NIST Handbook of Mathematical Functions, Cambridge Univ. Press, Cambridge, 2010.

Prudnikov, A. P., Brychkov, Yu. A., Marichev, O., Integrals and Series. Vol. 3. More Special Functions, (translated from the Russian by G. G. Gould), Gordon and Breach Sci. Publ., New York, 1990.

Rainville, E. D., Special Functions, Reprint of 1960 first edition, Chelsea Publishing Co., Bronx, N.Y., 1971.

Slater, L., Generalized Hypergeometric Functions, Cambridge Univ. Press, Cambridge, 1966.

Whipple, F. J. W., A group of generalized hypergeometric series: relations between 120 allied series of the type F2 h a,b,c e,f i, Proc. London Math. Soc. (2) 23 (1924), 104–114.

Whipple, F. J. W., Some transformations of generalized hypergeometric series, Proc. London Math. Soc. (2) 26 (1927), 257–272.




DOI: http://dx.doi.org/10.17951/a.2024.78.1.37-73
Date of publication: 2024-07-29 22:47:27
Date of submission: 2024-07-22 17:48:41


Statistics


Total abstract view - 221
Downloads (from 2020-06-17) - PDF - 135

Indicators



Refbacks

  • There are currently no refbacks.


Copyright (c) 2024 Thomas Ernst