Sur la dégénérescence de quelques formules de connexion pour les fonctions hypergéométriques de Gauss

摘要

It is well known that there exists an operation of limit that makes the degeneration of Gauss hypergeometric differential equation to the confluent hypergeometric differential equation. With this operation, we calculate the degeneration of connection formulas of Gauss hypergeometric function explicitely. This kind of degeneration connects the theory of analytic continuation of Gauss hypergeometric function with that of confluent hypergeometric function. In the case of the connection formula between 0 and $infty$ ( $S$ 2), we would like to calculate the limit of the formula: 1.4 $$F(alpha, beta, gamma, z/beta) = {frac{Gamma(gamma)Gamma(beta-alpha)}{Gamma(beta)Gamma(gamma-alpha)}}(e^{-pi i}z/beta)^{-alpha}F(alpha, 1 , +, alpha , -, gamma, 1,+,alpha , - ,beta, beta/z) +{frac{Gamma(gamma)Gamma(alpha-beta)}{Gamma(alpha)Gamma(gamma-beta)}}(e^{-pi i}z/beta)^{-beta}F(beta, 1+beta - gamma, 1+ beta -alpha , beta/z)$$ as $betarightarrowinfty$ . It is the problem that the right-hand side of (1.4) diverges if we take the limit $betarightarrowinfty$ for it directly. The problem will be resolved if we follow the following procedure: First, we replace $(e^{-pi i}Z/beta)^{-beta}F(beta , 1+beta - gamma ,1+beta - alpha ,beta /z)$ in the right-hand side of (1.4) by, after Kummer [4], the function $(e^{-pi i}z/beta)^{alpha -gamma}(1+e^{-pi i}z/beta)^{gamma-alpha-beta}F(1-alpha,gamma - alpha , beta-alpha+1,beta/z)$ . Next, we replace the two functions $F(alpha,1+alpha -gamma,1+alpha-beta,beta/z)$ and $F(1-alpha,gamma-alpha,beta-alpha+1,beta/z)$ by suitable Pochhammer integral representations. After these replacements, if we take the limit $betarightarrow e^{i theta}infty$ , then we have (Théorème 2.5) 2.9 $$F(alpha,gamma,z)={frac{Gamma(gamma)z^{-alpha}}{Gamma(gamma-alpha)Gamma(alpha)(e^{pi i alpha}-e^{-pi i alpha})}}int_{e^{i (theta-pi)}infty}^{(0 + )} e^{-upsilon}upsilon^{alpha-1}left(1+{frac{upsilon}{z}} right)^{gamma-alpha-1}dupsilon, +, {frac{Gamma(gamma)z^{alpha-gamma}e^z}{Gamma(alpha)Gamma(gamma-alpha)(e^{2pi i (gamma-alpha)}-1)}}int_{e^{i (theta-pi)}infty}^{(0 + )} e^{-upsilon}upsilon^{gamma-alpha-1}left(1-frac{upsilon}{z} right)^{alpha-1}dupsilon$$ (if $frac{1}{2}pi < theta < pi , text{or},pi < theta < frac{3}{2}pi$ ) under certain conditions for the parameters $alpha, beta, gamma$ and the independent variable z. A similar procedure works also for the case of the connection formula between 0 and 1 ( $S$ 3, Théorème 3.5).