The multiple hypergeometric generating function (1.3) below, due to H. M. Srivastava and J. P. Singhal [Acad. Roy. Belg. Bull. Cl. Sci. (5)58 (1972), 1238–1247], applies readily to deduce multilinear generating functions for thespecial Jacobi polynomialsPn(α-n,β)(x), Pn(α,β-n)(x) orPn(α-n,β-n)(x), the Laguerre polynomialsLn(α)(x), thebiorthogonal polynomialsZnα(x; k) of J.D.E. Konhauser [Pacific J. Math.21 (1967), 303–314], and so on, and indeed also for any suitable products of these polynomials. The present paper is motivated by the need for a multiple hypergeometric generating function, analogous to (1.3), which could apply to yield multilinear generating functions for theunrestricted Jacobi polynomialsPn(α,β)(x). Several interesting generalizations of the multiple hypergeometric generating function (1.3), and of its analogue (5.6) thus obtained, are given; many of these generalizations are shown to apply also to derive multilinear generating functions for the classical Hermite polynomialsHn(x) and for their various known generalizations considered, among others, by F. Brafman [Canad. J. Math.9 (1957), 180–187] and by H. W. Gould and A. T. Hopper [Duke Math. J.29 (1962), 51–63].
The multilinear generating functions (1.19), (1.22), (1.23), (1.25), (1.30), (3.3), (4.1), (4.2), (4.8), (5.5), (5.6), (6.3), (6.4) and (6.6) below are believed to be new.
(AMS) 1980 Mathematics Subject Classification
Primary 33A65 33A30 Secondary 42C15
This is a preview of subscription content, log in to check access.
Konhauser J. D. E.,Biorthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math.,21 (1967), 303–314.MATHMathSciNetGoogle Scholar
Lauricella G.,Sulle funzioni ipergeometriche a più variabili, Rend. Circ. Mat. Palermo,7 (1893), 111–158.CrossRefGoogle Scholar
Madhekar H. C., Thakare N. K.,Multilinear generating functions for Jacobi polynomials and for their two-variable generalizations, Indian J. Pure Appl. Math.,13 (1982), 711–716.MATHMathSciNetGoogle Scholar
Patil K. R., Thakare N. K.,Multilinear generating function for the Konhauser biorthogonal polynomial sets, SIAM J. Math. Anal.,9 (1978), 921–923.MATHCrossRefMathSciNetGoogle Scholar
Rainville E. D.,Special Functions, Macmillan, New York, 1960; Reprinted by Chelsea, Bronx, New York, 1971.MATHGoogle Scholar
Srivastava H. M., Daoust M. C.,Certain generalized Neumann expansions associated with the Kampé de Fériet function, Nederl. Akad. Wetensch. Proc. Ser. A,72=Indag. Math.,31 (1969), 449–457.MathSciNetGoogle Scholar
Srivastava H. M., Panda R.,Some analytic or asymptotic confluent expansions for functions of several variables, Math. Comput.,29 (1975), 1115–1128.MATHCrossRefMathSciNetGoogle Scholar
Srivastava H. M., Singhal J. P.,Some formulas involving the products of several Jacobi or Laguerre polynomials, Acad. Roy. Belg. Bull. Cl. Sci., (5),58 (1972), 1238–1247.MathSciNetGoogle Scholar