Formula for analytic continuation of the Kamp\'e de F\'eriet hypergeometric function

A. Hasanov ; T. K. Yuldashev

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, geometry, differential equations, Tome 217 (2022), pp. 97-106.

Résumé

We apply the method of Burchnall—Chaundy operators to the study of expansion formulas for the Kampé de Férriet hypergeometric function $F_{1:1;1}^{0:3;3} [x,y]$. Using the obtained operator identities, we derive 14 expansion formulas. A new group of Euler-type integral representations for the Kampé de Férriet hypergeometric function $F_{1:1;1}^{0:3;3} [x,y]$ is found and its analytic continuation is constructed.

Détail

Keywords: Kampé de Férriet hypergeometric function, Bourchnall–Chaundy operator, integral representation, expansion formula, analytic continuation.

Bibliographie

[1] Appell P., Kampé de Fériet J., Fonctions hypergeometriques et hyperspheriques. Polynomes d'Hermite, Gauthier-Villars, Paris, 1926

[2] Burchnall J. L., Chaundy T. W., “Expansions of Appell's double hypergeometric functions”, Quart. J. Math., 11 (1940), 249–270 | DOI | MR

[3] Burchnall J. L., Chaundy T. W., “Expansions of Appell's double hypergeometric functions, II”, Quart. J. Math., 12 (1941), 112–128 | DOI | MR

[4] Chaundy T. W., “Expansions of hypergeometric functions”, Quart. J. Math., 13 (1942), 159–171 | DOI | MR

[5] Erdelyi A., Magnus W., Oberhettinger F., Tricomi F. G., Higher transcendental functions. Vols. 1, 2, McGraw-Hill, New York, 1953 | MR

[6] Hasanov A., Srivastava H. M., Turaev M., “Decomposition formulas for some triple hypergeometric functions”, J. Math. Anal. Appl., 324 (2006), 955–969 | DOI | MR

[7] Lauricella G., “Sulle funzioni ipergeometriche a piu variabili”, Rend. Circ. Mat. Palermo., 7 (1893), 111–158 | DOI

[8] Marichev O. I., Handbook of Integral Transforms of Higher Transcendental Functions: Theory and Algorithmic Tables, Wiley, New York, 1982 | MR

[9] Poole E. G., Introduction to the Theory of Linear Differential Equations, Oxford Univ. Press, Oxford, 1936

[10] Pandey R. C., “On the expansions of hypergeometric functions”, Agra Univ. J. Res. Sci., 12 (1963), 159–169

[11] Paris R. B., Kaminski D., Asymptotics and Mellin–Barnes Integrals, Cambridge Univ. Press, Cambridge, 2001 | MR

[12] Ragab F. M., “Expansions of Kampe de Feriet's double hypergeometric function of higher order”, J. Reine Angew. Math., 212 (1963), 113–119 | DOI | MR

[13] Singhal J. P., Bhati S. S., “Certain expansions associated with hypergeometric functions of variables”, Glasnik Mat. Ser., 3:11 (1976), 239–245 | MR

[14] Srivastava H. M., “Hypergeometric functions of three variables”, Ganita., 15 (1964), 97–108 | MR

[15] Srivastava H. M., “Some integrals representing triple hypergeometric functions”, Rend. Circ. Mat. Palermo. Ser. 2., 16 (1967), 99–115 | DOI | MR

[16] Niukkanen A. W., “Generalised hypergeometric series arising in physical and quantum chemical applications”, J. Phys. A: Math. Gen., 16 (1983), 1813–1825 | DOI | MR

[17] Srivastava H. M., “A class of generalized multiple hypergeometric series in physical and quantum chemical applications”, J. Phys. A: Math. Gen., 18 (1985), 227–234 | DOI | MR

[18] Srivastava H. M., Karlsson P. W., Multiple Gaussian Hypergeometric Series, Wiley, New York, 1985 | MR

[19] Srivastava H. M., Manocha H., A Treatise on Generating Functions, Wiley, New York, 1984 | MR

[20] Turaev M., “Decomposition formulas for Srivastava's hypergeometric function on Saran functions”, J. Comput. Appl. Math., 233 (2009), 842–846 | DOI | MR

[21] Verma A., “Expansions involving hypergeometric functions of two variables”, Math. Comput., 20 (1966), 590–596 | DOI | MR

[22] Hasanov A., Yuldashev T. K., “Analytic continuation formulas for the hypergeometric functions in three variables of second order”, Lobachevskii J. Math., 43:2 (2022), 386–393 | DOI | MR