### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 074, 22 pages      arXiv:1504.08144      https://doi.org/10.3842/SIGMA.2015.074
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

### Fractional Integral and Generalized Stieltjes Transforms for Hypergeometric Functions as Transmutation Operators

Tom H. Koornwinder
Korteweg-de Vries Institute, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands

Received April 29, 2015, in final form September 14, 2015; Published online September 20, 2015

Abstract
For each of the eight $n$-th derivative parameter changing formulas for Gauss hypergeometric functions a corresponding fractional integration formula is given. For both types of formulas the differential or integral operator is intertwining between two actions of the hypergeometric differential operator (for two sets of parameters): a so-called transmutation property. This leads to eight fractional integration formulas and four generalized Stieltjes transform formulas for each of the six different explicit solutions of the hypergeometric differential equation, by letting the transforms act on the solutions. By specialization two Euler type integral representations for each of the six solutions are obtained.

Key words: Gauss hypergeometric function; Euler integral representation; fractional integral transform; Stieltjes transform; transmutation formula.

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References

1. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
2. Askey R., Orthogonal polynomials and special functions, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975.
3. Askey R., Fitch J., Integral representations for Jacobi polynomials and some applications, J. Math. Anal. Appl. 26 (1969), 411-437.
4. Bateman H., The solution of linear differential equations by means of definite integrals, Trans. Cambridge Philos. Soc. 21 (1909), 171-196.
5. Camporesi R., The biradial Paley-Wiener theorem for the Helgason Fourier transform on Damek-Ricci spaces, J. Funct. Anal. 267 (2014), 428-451.
6. Dereziński J., Hypergeometric type functions and their symmetries, Ann. Henri Poincaré 15 (2014), 1569-1653, arXiv:1305.3113.
7. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vol. I, Mc-Graw Hill, New York, 1953.
8. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vol. II, Mc-Graw Hill, New York, 1953.
9. Flanders H., Differentiation under the integral sign, Amer. Math. Monthly 80 (1973), 615-627, Correction, Amer. Math. Monthly 81 (1974), 145.
10. Karp D., Prilepkina E., Generalized Stieltjes functions and their exact order, J. Class. Anal. 1 (2012), 53-74.
11. Karp D., Prilepkina E., Hypergeometric functions as generalized Stieltjes transforms, J. Math. Anal. Appl. 393 (2012), 348-359, arXiv:1112.5769.
12. Karp D., Sitnik S.M., Inequalities and monotonicity of ratios for generalized hypergeometric function, J. Approx. Theory 161 (2009), 337-352, math.CA/0703084.
13. Kodavanji S., Rathie A.K., Paris R.B., A derivation of two transformation formulas contiguous to that of Kummer's second theorem via a differential equation approach, arXiv:1501.06173.
14. Letnikov A.V., Research related to the theory of integrals of the form $\int_0^x (x-u)^{p-1} f(u) du$. Chapter III. Application to the integration of certain differential equations, Mat. Sb. 7 (1874), 111-205 (in Russian).
15. Lions J.L., Opérateurs de Delsarte et problèmes mixtes, Bull. Soc. Math. France 84 (1956), 9-95.
16. Miller K.S., Ross B., An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993.
17. Miller Jr. W., Lie theory and generalizations of the hypergeometric functions, SIAM J. Appl. Math. 25 (1973), 226-235.
18. Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W. (Editors), NIST handbook of mathematical functions, U.S. Department of Commerce National Institute of Standards and Technology, Washington, DC, Cambridge University Press, Cambridge, 2010, available at http://dlmf.nist.gov.
19. Rainville E.D., Special functions, The Macmillan Co., New York, 1960.
20. Saito M., Symmetry algebras of normal ${\mathcal A}$-hypergeometric systems, Hokkaido Math. J. 25 (1996), 591-619.
21. Sitnik S.M., Transmutations and applications: a survey, arXiv:1012.3741 (in Russian).
22. Sitnik S.M., Buschman-Erdelyi transmutations, classification and applications, arXiv:1304.2114.
23. Sostak R.Ya., Aleksei Vasilevic Letnikov, Istor.-Mat. Issled. 5 (1952), 167-238 (in Russian).
24. Swathi M., Rathie A.K., Paris R.B., A derivation of two quadratic transformations contiguous to that of Gauss via a differential equation approach, arXiv:1411.5262.
25. Szegő G., Orthogonal polynomials, American Mathematical Society, Colloquium Publications, Vol. 23, 4th ed., Amer. Math. Soc., Providence, R.I., 1975.
26. Widder D.V., The Stieltjes transform, Trans. Amer. Math. Soc. 43 (1938), 7-60.