where is a generalized hypergeometric function and is the gamma function (Bailey 1935, p. 16; Koepf 1998, p. 32).
The generalized hypergeometric function is given by a hypergeometricseries, i.e., a series for which the ratio of successive terms can be written(1)(The factor of in the denominator is present for historical reasons of notation.) The resulting generalized hypergeometric function is written(2)(3)where is the Pochhammer symbol or rising factorial(4)A generalized hypergeometric function therefore has parameters of type 1 and parameters of type 2.A number of generalized hypergeometric functions has special names. is called a confluent hypergeometric limit function, and is implemented in the Wolfram Language as Hypergeometric0F1[b, z]. (also denoted ) is called a confluent hypergeometric function of the first kind, and is implemented in the Wolfram Language as Hypergeometric1F1[a, b, z]. The function is often called "the" hypergeometric function or Gauss's hypergeometric function, and is implemented in the Wolfram..
Thomae's theorem, also called Thomae's transformation, is the generalizedhypergeometric function identity(1)where is the gamma function, is a generalized hypergeometric function,(2)and (Bailey 1935, p. 14). It is a generalization of Dixon's theorem (Slater 1966, p. 52).An equivalent formulation is given by(3)(Hardy 1999, p. 104). The symmetry of this form was used by Ramanujan in his proof of the identity, which is essentially the same as Thomae's. Interestingly, this is one of the few cases in which Ramanujan gives an explicit proof of one of his propositions (Hardy 1999, p. 104).A special case of the theorem is given by(4)(J. Sondow, pers. comm., May 25, 2003).
Lauricella functions are generalizations of the Gauss hypergeometric functions to multiple variables. Four such generalizations were investigated by Lauricella (1893), and more fully by Appell and Kampé de Fériet (1926, p. 117). Let be the number of variables, then the Lauricella functions are defined byIf , then these functions reduce to the Appell hypergeometric functions , , , and , respectively. If , all four become the Gauss hypergeometric function (Exton 1978, p. 29).
A special function generalizes the generalized hypergeometric function to two variables and includes the Appell hypergeometric function as a special case. The Kampe de Feriet function can represent derivatives of generalized hypergeometric functions with respect to their parameters, as well as indefinite integrals of two and three Meijer G-functions. Exton and Krupnikov (1998) have derived a large collection of formulas involving this function.Kampé de Fériet functions are written in the notation(1)Special cases include(2)(3)for and , where is the incomplete elliptic integral of the second kind and is the incomplete elliptic integral of the first kind, as well as(4)for , where is the incomplete elliptic integral of the third kind (Exton and Krupnikov 1998, p. 1). Additional identities are given by(5)(6)(7)(Exton and Krupnikov 1998, p. 3)...
The 34 distinct convergent hypergeometric series of order two enumerated by Horn (1931) and corrected by Borngässer (1933). There are 14 complete series for which :(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(of which , , , and are precisely Appell hypergeometric functions), and 20 confluent series with , , and not both 2:(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32)(33)(34)(Erdélyi et al. 1981, pp. 224-226; Srivastava and Karlsson 1985, pp. 24-26). Here, the sums are taken over nonnegative integers and .Note that , , and as defined by Erdélyi et al. (1981) are erroneous; the correct formulas given above may be found in Srivastava and Karlsson (1985, pp. 25-26).
A formal extension of the hypergeometric function to two variables, resulting in four kinds of functions (Appell 1925; Picard 1880ab, 1881; Goursat 1882; Whittaker and Watson 1990, Ex. 22, p. 300),(1)(2)(3)(4)These double series are absolutely convergent for(5)Appell defined the functions in 1880 and they were subsequently studied by Picard in 1881. The functions , , and can be expressed in terms of double integrals as(6)(7)(8)(Bailey 1934, pp. 76-77). There appears to be no simple integral representation of this type for the function (Bailey 1934, p. 77).The function can also be expressed by the simple integral(9)(Bailey 1934, p. 77), for and .The Appell functions are special cases of the Kampé de Fériet function, and are the first four in the set of Horn functions. The function is implemented in the Wolfram Language as AppellF1[a, b1, b2, c, x, y].For general complex parameters, the function..