Double Chebyshev series for hypergeometric functions of two variables
Treść / Zawartość
We give formulas for the coefficients of a double Chebyshev series for a hypergeometric function of two variables x and y. We express these coefficients in terms of other hypergeometric functions of two variables. In particular, for hypergeometric functions expressed in terms of corresponding hypergeometric functions of one variable with an argument of the form x+y, the Chebyshev coefficients are values of another hypergeometric function of one variable. In Section 1 we give basic information on double Chebyshev series. Their many numerical applications are well known. We also develop algorithms for computing partial sums of series of this type. Basu gave a generalization of Clenshaw's algorithm for summing a single series to the case of double Chebyshev series." In Section 2 we give basic information on hypergeometric functions of two variables. In Section 3 we give formulas for the coefficients of a double Chebyshev series of any hypergeometric function of two variables and a simplified version of these models for particular forms of these functions. In Section 4 we prove two of these formulas. The proofs of the other formulas are analogous and are therefore omitted. This study was carried out within an interdisciplinary program of 'Mathematical theories and their applications'. Some of the results have been published earlier [see, e.g., the author, Bull. Acad. Polon. Sci. Ser. Sci. Math. 23 (1975), no. 10, 1107–1111; MR0399532]."