Some transformations of particular hypergeometric series
Abstract Generated abstract
This paper derives several transformation formulas for special hypergeometric series and discusses their computational use. It first transforms a finite polynomial in powers of z into a sum involving powers of 1 minus z and ratios of 1 minus z to z, yielding particular summation identities. It then presents a representation of F(alpha, 1, gamma, z) as a finite polynomial in z divided by 1 minus z plus a shifted hypergeometric function F(alpha, 1, gamma plus n, z), with comments on convergence and error estimation via continued fractions. Finally, it gives parameter-shift relations for F(alpha, beta, gamma, 1), including a connection with the beta function, and indicates their use for accurate computation.
Full Text
SOME TRANSFORMATIONS OF PARTICULAR HYPERGEOMETRIC SERIES
V. E. KORNILOV
(Presented by the Department of Higher Mathematics)
In this article the following transformations are considered:
a. The transformation of a polynomial in integral powers of \(z\) into a sum of two polynomials in powers of \(1-z\) and \((1-z):z\).
b. For the computation of the function \(F(\alpha,1;\gamma;z)\), its representation is proposed as the sum of a polynomial in powers of \(z:(1-z)\) and the function \(F(\alpha,1;\gamma+n;z)\).
c. Relations are given between the function \(F(\alpha,\beta;\gamma;1)\) and similar functions whose parameters have been increased by an integer.
- On the basis of the equalities ([1], p. 17, (4)—(6)) we have
\[ \sum_{k=0}^{n-1}\frac{(\alpha)_k}{(\gamma)_k}z^k + \frac{(\alpha)_n}{(\gamma)_n}z^n F(\alpha+n,1;\gamma+n;z) = \]
\[ =(1-\gamma)\sum_{k=0}^{n-1} \frac{(\alpha)_k}{(\alpha-\gamma+1)_{k+1}}(1-z)^k + \frac{(\alpha)_n(1-z)^n}{(\alpha-\gamma+1)_n} \left[ -\sum_{k=0}^{n-1} \frac{(\gamma-\alpha-n)_k}{(\gamma)_k}\, \frac{z^k}{(z-1)^{k+1}} +\right. \]
\[ \left. + \frac{(\alpha-\gamma+1)_n}{(\gamma)_n} \left(\frac{z}{1-z}\right)^n F(\alpha+n,1;\gamma+n;z) \right]. \]
After expanding the square brackets and collecting like terms, and also after summing the second sum in reverse order, we finally obtain:
\[ T(z)= \sum_{k=0}^{n-1}\frac{(\alpha)_k}{(\gamma)_k}t^k \equiv \sum_{k=0}^{n-1} \frac{(\alpha)_k(1-\gamma)}{(\alpha-\gamma+1)_{k+1}} (1-z)^k + \]
\[ + \frac{(\alpha)_n z^{\,n-1}}{(\gamma)_{n-1}} \sum_{k=0}^{n-1} \frac{(\gamma+n-1-k)_k}{(\alpha-\gamma+1)_{k+1}} \left(\frac{1-z}{z}\right)^k . \tag{1} \]
In particular cases, for \(z=1\) and \(z=1,\ \gamma=1\), we have
\[ \sum_{k=0}^{n-1}\frac{(\alpha)_k}{(\gamma)_k} = \frac{(\alpha)_n-(\gamma-1)_n}{(\gamma)_{n-1}(\alpha-\gamma+1)} ;\qquad \sum_{k=0}^{n-1}\frac{(\alpha)_k}{k!} = \frac{(\alpha+1)_{n-1}}{(n-1)!}. \tag{2} \]
- The identity ([1], p. 17, (6))
\[ F(\alpha, 1; \gamma; z) = \frac{1}{1-z} \sum_{k=0}^{n-1} \frac{(\gamma-\alpha)_k}{(\gamma)_k} \left(\frac{z}{z-1}\right)^k + \frac{(\gamma-\alpha)_n}{(\gamma)_n} \left(\frac{z}{z-1}\right)^n F(\alpha, 1; \gamma+n; z) \tag{3} \]
can be successfully applied to the computation of the function \(F(\alpha, 1; \gamma; z)\), since the polynomial situated on the right-hand side of identity (3) gives a good approximation for any \(|z|>0\), \(\operatorname{Re} z \leqslant 0\), and \(\gamma>\alpha\). The function situated on the right-hand side of identity (3) is expanded into a continued fraction, which also gives good convergence. Moreover, the estimate of the remainder term of a convergent fraction is, in modulus, less than the modulus of the difference of two nearest convergents in the domain \(\operatorname{Re} z \leqslant 0\).
With the aid of the series \(F(\alpha, 1; \gamma; z)\) one can compute the elementary functions \(\operatorname{arctg} z\), \(\ln(1+z)\), \((1+z)^{-\alpha}\), and other functions.
- After successive application of the elementary relations between contiguous hypergeometric functions ([2], p. 111) to the function \(F \equiv F(\alpha, \beta; \gamma; 1)\), we obtain the following identities:
\[ F = \frac{(\gamma-\alpha)_n(\gamma-\beta)_n} {(\gamma)_n(\gamma-\alpha-\beta)_n} F(\alpha, \beta; \gamma+n; 1); \tag{4} \]
\[ F = \frac{(\gamma-\alpha)_n} {(\gamma-\alpha-\beta)_n} F(\alpha-n, \beta; \gamma; 1); \tag{5} \]
\[ F = \frac{(\gamma-\alpha)_{n+m}(\gamma-\beta)_n} {(\gamma)_n(\gamma-\alpha-\beta)_{n+m}} F(\alpha-m, \beta; \gamma+n; 1); \tag{6} \]
\[ F = \frac{(\gamma-\alpha)_{n+m}(\gamma-\beta)_{n+k}} {(\gamma)_n(\gamma-\alpha-\beta)_{n+m+k}} F(\alpha-m, \beta-k; \gamma+n; 1). \tag{7} \]
A particular case of the function \(F(\alpha, \beta; \gamma; 1)\) is the beta function
\[ B(x,y)=\frac{1}{x}F(1-y, x; x+1; 1). \tag{8} \]
The application of formulas (4)—(7) to the function \(F(\alpha,\beta;\gamma;1)\) and the representation by convergent fractions of the functions situated on the right-hand sides of these identities make it possible to compute it with high accuracy and with the least expenditure of computation time.
References
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V. E. Kornilov. Separation of the algebraic part of integrals of binomial differentials. Izvestiya TPI, vol. 131, pp. 17–20, 1965.
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G. Bateman and A. Erdélyi. Higher Transcendental Functions (Hypergeometric Function, Legendre Functions). Moscow, Fizmatgiz, 1965.