An asymptotic expansion for a ratio of products of gamma functions

An asymptotic expansion of a ratio of products of gamma functions is derived. It generalizes a formula which was stated by Dingle, first proved by Paris, and recently reconsidered by Olver.

The hypergeometric series appears as one solution of the Gaussian (or hypergeometric) differential equation, which is characterized by its three regular singular points at z = 0, 1, ∞. The local series solutions at 0 and 1 of this differential equation are connected by the continuation formula [1] 1 Here we want to show that Eq. (1) implies an interesting asymptotic expansion for a ratio of products of gamma functions, of which only a special case was known before.
By applying the method of Darboux [4,8] to (1), we derive in Sec. 2 the formula in question. The behaviour of this and a related formula is discussed in Sec. 3 and illustrated by a few numerical examples.

Derivation of an asymptotic expansion for a ratio of products of gamma functions
It is well-known that the late coefficients of a Taylor series expansion contain information about the nearest singular point of the expanded function [3]. In this respect we want to analyze the continuation formula (1), in which then only the second, at z = 1 singular term R is relevant, which may be written as By means of the binomial theorem in its hypergeometric-series-form , we may expand the power factor Interchanging then the order of the summations and simplifying by means of the reflection formula of the gamma function, we arrive at This is to be compared with the left-hand side L of (1), which is Comparison of the coefficients of these two power series, which according to Darboux [4] and Schäfke and Schmidt [8] should agree asymptotically as n → ∞, then yields

By means of
and the reflection formula of the gamma function, the relevant formula (2) may also be written as (3) The asymptotic expansion for a ratio of products of gamma functions in this form (3) or the other (2) seems to be new. It is only the special case when c = 1 which is known. This special case was stated by Dingle [2], first proved by Paris [7], and reconsidered recently by Olver [5], who has found a simple direct proof. His proof, as well as the proof of Paris, can be adapted easily to the more general case when c is different from 1 . Still another proof is available [6] which includes an integral representation of the remainder term. Our derivation of Eq. (2) or (3) is significantly different from all the earlier proofs of the case when c = 1.

Discussion and numerical examples
We now want to discuss our result in the form (3). First we observe that the substitution c → a + b − c leads to the related formula Which of (3) or (4) is more advantageous numerically depends on the values of the parameters, and in this respect the two formulas complement each other. Table 1 shows an example with a set of parameters for which (3) gives more accurate values than (4), while Table 2 contains an example for which (4) is superior to (3). For finite n and M → ∞ the series on the right-hand side of (3) converges if Re(1 − c − n) > 0. The same is true for (4) Then, in both cases, the Gaussian summation formula yields which, by means of the reflection formula of the gamma function, is seen to be equal to Otherwise (2) -(4) are divergent asymptotic expansions as n → ∞. Although in our derivation n is a sufficiently large positive integer, the asymptotic expansions (2) -(4) are expected to be valid in a certain sector of the complex n -plane, and in fact, the proofs of Paris [7] and of Olver [6] apply to complex values of n.
If the series in (3) or (4) converge, their sums are equal to (5), which generally (if neither c − a nor c − b is equal to an integer ) is different from the left-hand side of (3) or (4). Therefore (3) and (4) can be valid only in the half-planes in which the series do not converge. This means that (3) is an asymptotic expansion as n → ∞ in the half-plane Re(c−1+n) ≥ 0, and (4) is an asymptotic expansion as n → ∞ in the half-plane Re(a+b−c−1+n) ≥ 0. Otherwise the series on the right-hand sides represent a different function, namely (5).
A few numerical examples may serve for demonstration of these facts. In Table 3 , the series converge to (5) for n = 10 , and therefore (3) and (4) are not valid. For n = 20, on the other hand, the series diverge and so (3) and (4) hold. The transition between the two regions is at the line Re(n) = 12.4 in case of (3) or Re(n) = 12.5 in case of (4). In Table 4, we see convergence for n = −15 and divergence for n = −5, the transition between the two regions being at the line Re(n) = −10.4 in case of (3) or Re(n) = −10.5 in case of (4) .   Table 3: Values of the right-hand sides of (3) and (4) for the parameters a = −11.7, b = −11.2, c = −11.4 .
M right-hand side of (3) right-hand side of(4) n  Table 4: Values of the right-hand sides of (3) or (4) for the parameters a = 11.7, b = 11.2, c = 11.4.