SOME FAMILIES OF GENERALIZED ELLIPTIC-TYPE INTEGRALS AND THE ASSOCIATED MULTIPLE HYPERGEOMETRIC FUNCTIONS

Motivated essentially by the fact that several interesting families of generalized elliptic-type integrals and many definite integrals of such families with respect to their modulus (or complementary modulus) are known to arise naturally in various physical contexts, we present a systematic account of the theory of these families of elliptic-type integrals in a unified and generalized manner. We also point out relevant connections of the general theory, which is discussed in this paper, with the results in numerous earlier publications on the theory and applications of the aforementioned families of elliptictype integrals. Several classes of hypergeometric functions of one, two and more variables are observed to emerge in our unified presentation of these families of elliptic-type integrals.