Linearization relations for the generalized Bedient polynomials of the first and second kinds via their integral representations

Lin, Shy-Der ; Liu, Shuoh-Jung ; Lu, Han-Chun ; Srivastava, Hari Mohan

Czechoslovak Mathematical Journal, Tome 63 (2013) no. 4, pp. 969-987

Abstract (VO)

The main object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations. By suitably specializing our main results, the corresponding integral representations are deduced for such familiar classes of hypergeometric polynomials as (for example) the generalized Bedient polynomials of the first and second kinds. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials.

Détail

DOI : 10.1007/s10587-013-0065-6

Classification : 33C45, 33C65, 42C05
Keywords: hypergeometric function; hypergeometric polynomial; Srivastava polynomial; Bedient polynomial; generalized Bedient polynomial of the first and second kinds; multiple integral representation; Gamma function; Eulerian beta integral linearization relationship; Pochhammer symbol; shifted factorial

Bibliographie

[1] Appell, P., Fériet, J. Kampé de: Fonctions hypergéométriques et hypersphériques. Polynomes d'Hermite. Gauthier-Villars Paris (1926).

[2] Bedient, P. E.: Polynomials Related to Appell Functions of Two Variables. Ph.D. Thesis University of Michigan (1959). | MR

[3] Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G.: Higher Transcendental Functions Vol. I. Bateman Manuscript Project McGraw-Hill Book Co. New York (1953). | Zbl

[4] González, B., Matera, J., Srivastava, H. M.: Some $q$-generating functions and associated generalized hypergeometric polynomials. Math. Comput. Modelling 34 (2001), 133-175. | DOI | MR

[5] Khan, M. A., Khan, A. H., Singh, M.: Integral representations for the product of Krawtchouk, Meixner, Charlier and Gottlieb polynomials. Int. J. Math. Anal., Ruse 5 (2011), 199-206. | MR | Zbl

[6] Lin, S.-D., Chao, Y.-S., Srivastava, H. M.: Some families of hypergeometric polynomials and associated integral representations. J. Math. Anal. Appl. 294 (2004), 399-411. | DOI | MR | Zbl

[7] Lin, S.-D., Liu, S.-J., Lu, H.-C., Srivastava, H. M.: Integral representations for the generalized Bedient polynomials and the generalized Cesàro polynomials. Appl. Math. Comput. 218 (2011), 1330-1341. | DOI | MR | Zbl

[8] Lin, S.-D., Liu, S.-J., Srivastava, H. M.: Some families of hypergeometric polynomials and associated multiple integral representations. Integral Transforms Spec. Funct. 22 (2011), 403-414. | DOI | MR | Zbl

[9] Lin, S.-D., Srivastava, H. M., Wang, P.-Y.: Some families of hypergeometric transformations and generating relations. Math. Comput. Modelling 36 (2002), 445-459. | DOI | MR | Zbl

[10] Liu, S.-J., Chyan, C.-J., Lu, H.-C., Srivastava, H. M.: Multiple integral representations for some families of hypergeometric and other polynomials. Math. Comput. Modelling 54 (2011), 1420-1427. | DOI | MR | Zbl

[11] Magnus, W., Oberhettinger, F., Soni, R. P.: Formulas and Theorems for the Special Functions of Mathematical Physics. 3rd enlarged ed., Die Grundlehren der mathematischen Wissenschaften 52 Springer, Berlin (1966). | MR | Zbl

[12] Srivastava, H. M.: A contour integral involving Fox's $H$-function. Indian J. Math. 14 (1972), 1-6. | MR | Zbl

[13] Srivastava, H. M., Joshi, C. M.: Integral representation for the product of a class of generalized hypergeometric polynomials. Acad. R. Belg., Bull. Cl. Sci., V. Ser. 60 (1974), 919-926. | MR | Zbl

[14] Srivastava, H. M., Lin, S.-D., Liu, S.-J., Lu, H.-C.: Integral representations for the Lagrange polynomials, Shively's pseudo-Laguerre polynomials, and the generalized Bessel polynomials. Russ. J. Math. Phys. 19 (2012), 121-130. | DOI | MR | Zbl

[15] Srivastava, H. M., Manocha, H. L.: A Treatise on Generating Functions. Ellis Horwood Series in Mathematics and Its Applications Ellis Horwood Limited, Chichester (1984). | MR | Zbl

[16] Srivastava, H. M., Özarslan, M. A., Kaanoğlu, C.: Some families of generating functions for a certain class of three-variable polynomials. Integral Transforms Spec. Funct. 21 (2010), 885-896. | DOI | MR | Zbl

[17] Srivastava, H. M., Panda, R.: An integral representation for the product of two Jacobi polynomials. J. Lond. Math. Soc., II. Ser. 12 (1976), 419-425. | DOI | MR | Zbl

[18] Szegö, G.: Orthogonal Polynomials. 4th ed. American Mathematical Society Colloquium Publications Vol. 23 AMS, Providence (1975). | MR | Zbl

[19] Whittaker, E. T., Watson, G. N.: A Course of Modern Analysis. An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions, Repr. of the 4th ed. 1927 Cambridge University Press, Cambridge (1996). | MR | Zbl