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Gaussian Hypergeometric Series and Its Generalizations
(20)
An obvious special case of (20) occurs when the numerator parameter a or b is a non-positive integer -n, say. We thus have the summation formula
(21)
which incidentally is equivalent to Vandermonde’s convolution theorem:
(22)
λ and µ being any complex numbers.
For a number of summation theorems for the hypergeometric series (17) when z takes on other special values, see Bailey (1935a, pp. 9-11), Erdelyi et al (1953, pp. 104-105), Slater (1966, p. 243), and Luke (1975, pp. 271-273).
Generalized Hypergeometric Series
In the Gaussian hypergeometric series F(a, b; c; z) there are two numerator parameters a, b, and one denominator parameter c. A natural generalization of this series is accomplished by introducing any arbitrary number of numerator and denominator parameters. The resulting series
(23)
where is the Pochhammer symbol defined by (2), is known as the generalized Gauss series, or simply, the generalized hypergeometric series. Here p and q are positive integers or zero (interpreting an empty product as 1), and we assume that the variable z, the numerator parameters and the denominator parameters take on complex values, provided that
(24)
With this notation the Gaussian series F(a, b; c; z) is also denoted by 2F1,(a, b; c; z) as in (20) and (21) above, or by
Supposing that none of the numerator parameters is zero or a negative integer (otherwise the question of convergence will not arise), and with the usual restriction (24),the pFq series in (23)
converges for
converges for |z|<1 if p = q + 1, and
diverges for all z, z≠ 0, if p > q +1.
Furthermore, if we set
(25)
it is known that the pFq series, with p = q + 1, is
absolutely convergent for |z| = 1 if Re(ω) > 0,
conditionally convergent for |z| = 1, z ≠1, if — 1 < Re(ω) 0, and
divergent for |z| = 1 if Re(ω) -1.
An important special case of the series (23) is the Kummerian hypergeometric series 1F1(a;c; z) in which case p = q =1. Since
(26)
for bounded z and n = 0, 1, 2,... , we have
In view of the principle of confluence involved in (27), Rummer’s series 1F1(a;c; z) is also called the confluent hypergeometric series.
A fairly wide variety of special functions can be represented in terms of generalized hypergeometric series. We list below the hypergeometric representations of some of the elementary functions, and refer the reader to Srivastava and Kashyap (1982, pp. 22-23, 31-32, and 36-37) for such representations of many higher transcendental functions.
ez = 0F0(-;-;z),
(1-z)-a=1F0(a;-; z),
ln(l + z) = z 2F1(1, 1; 2; —z),
A Further Generalization of pFq
An interesting further generalization of the series pFq is due to Fox (1928) and Wright (1935, 1940) who studied the asymptotic expansion of the generalized hypergeometric function defined by
where the coefficients A1 , ... ,Ap and Bl , ... , Bq are positive real numbers such that
By comparing the definitions (23) and (38), we have
HYPERGEOMETRIC SERIES IN TWO VARIABLES
The enormous success of the theory of hypergeometric series in a single variable has stimulated the development of a corresponding theory in two and more variables. In this section we present a brief account of Appell series and their confluent cases, and indicate various generalizations of these series in two variables.
Appell Series
Consider the product of two Gaussian series, viz
This double series, in itself, yields nothing new, but by replacing one or more of the three pairs of products
(a)m(a')n , (b)m(b')n , (c)m(c')n
by the corresponding expressions
(a)m+n , (b)m+n , (c)m+n ,
we are led to five distinct possibilities of getting new double series. One such possibility, however, gives us the double series
which is simply the Gaussian series 2F1(a, b; c; x + y).
The remaining four possibilities lead us to the four double hypergeometric series (known as Appell series), which are defined below [Appell (1880, p. 296, Equations (1))]:
max{|x|,|y|}<1;
|x|+|y|<1;
max{|x|,|y|}<1;
here, as usual, the denominator parameters c and c' are neither zero nor a negative integer. The standard work on the theory of Appell series is the monograph by Appell and Kampe de Feriet (1926), which contains an extensive bibliography of all relevant papers up to 1926 (by, for example, L. Pochhammer, J. Нот, Ё. Picard, and Ё. Goursat). See Erd61yi et at. (1953, pp. 222-245) for a review of the subsequent work on the subject; see also Bailey (1935a, Chapter 9), Slater (1966, Chapter 8), and Exton (1976b, pp. 23-28).
Horn Series
Horn (1931) defined the following ten hypergeometric series in two variables and denoted them by G1 G2, G3, H1, ... , H7; he thus completed the set of all possible sccond-ordert (complete) hypergeometric series in two variables [Appell and Kampe de Feriet (1926, p. 143 etseq.); see also Erdelyi et al. (1953, pp. 224-228)].
C
t See Section 2.1 for a detailed description of the order classification for a multiple hypergeometric series.
Precise regions of convergence for these series are found in Section 2.2.
onfluent Hypergeometric Scries in Two Variables
Seven confluent forms of the four Appeli series were defined by Humbert (1920-21), and he denoted these confluent hypergeometric series in two variables by
In addition, there exist 13 confluent forms of the Horn series, which are denoted by [Horn (1931); Borngasser (1933)]
Г„ Г2, H,, ... , H„.
T
(16) Ф^а, P; у; x, у] =
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