Asymptotic analysis of a Selberg-type integral via hypergeometrics

C. Krattenthaler^{†}

Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, A-1090 Vienna, Austria.

                 WWW: http://www.mat.univie.ac.at/~kratt

Abstract. We show how to determine the asymptotics of a certain Selberg-type integral by means of tools available in the theory of (generalised) hypergeometric series. This provides an alternative derivation of a result of Carré, Deneufchâtel, Luque and Vivo [arχiv:1003.5996].

In [2], Carré, Deneufchâtel, Luque and Vivo consider the Selberg-type integral

S_{k}(a, b) = \frac{1}{N!} ∫_{[0,1]^{N}} x_{1}^{k}( ∏_{1≤i<j≤N}(x_{i} − x_{j})^{2})( ∏_{i=1}^{N} x_{i}^{a−1}(1 − x_{i})^{b−1} dx_{i}),

and they determine its asymptotic behaviour when N, a, b all tend to infinity so that a ∼ a_{1}N and b ∼ b_{1}N, where a_{1} and b_{1} are given non-negative real numbers. The reader is referred to the introduction of [2] for information on motivation from random matrix theory connected to random scattering theory to investigate this question.

It should be noted that S_{0}(a, b) is a Selberg integral, which can be evaluated in a product/quotient of gamma functions (cf. [3]). This being the case, the asymptotics of S_{0}(a, b) is easily determined by means of known asymptotic formulae for the Barnes Gfunction (see [7]). Thus, it suffices to consider the quotient

J_{k} = \frac{S_{k}(a, b)}{S_{0}(a, b)}

(this quotient is denoted by I_{k}/N in [2]) and determine its asymptotic behaviour. By (now) classical identities in the theory of symmetric functions, it is shown in [2, Cor. II.3] that

J_{k} = \frac{1}{N · k!} ∑_{i=0}^{k−1}(−1)^{i}\binom{k − 1}{i}\frac{(N − i)_{k} (a + N − i − 1)_{k}}{(a + b + 2N − i − 2)_{k}} ,  (1)

hypergeometric series, transformation formula. bilistic Number Theory”.

2000 Mathematics Subject Classification. Primary 33C20; Secondary 33A15 33C52 60B20 82B05.

Key words and phrases. Selberg-type integral, hypergeometric series, contiguous relation, balanced

†Research partially supported by the Austrian Science Foundation FWF, grants Z130-N13 and S9607-N13, the latter in the framework of the National Research Network “Analytic Combinatorics and Proba-

where (α)_{m} is the usual Pochhammer symbol defined by (α)_{m} = α(α + 1) ⋯ (α + m − 1) for m ≥ 1, and (α)_{0} = 1. In view of this rather simple expression, which, after substitution of a by a_{1}N and of b by b_{1}N, is a rational function in N, the problem of determining the asymptotic behaviour of the resulting expression looks innocent. However, inspection of the expression (1) shows that this is not so innocent at all: the summands are all of the order of magnitude N^{k} for large N, while (see Theorem and Corollary below), in fact, the expression in (1) has a finite limit as N → ∞ in the described situation. Hence, what happens here is a large amount of cancellation of leading asymptotic terms. This is what one has to get under control, and this constitutes the challenge in determining this limit.

To solve the problem, Carré, Deneufchâtel, Luque and Vivo develop a difference operator calculus in [2], which is of interest in its own right.

The purpose of the present note is to demonstrate that the tools for solving this problem are readily available in the literature on (generalised) hypergeometric series, and it is clear that they may serve as well to solve problems of similar type.

Namely, using the standard hypergeometric notation

pF_{q}[ \begin{matrix}a_{1}, …, a_{p} \\ b_{1}, …, b_{q} \end{matrix}; z] = ∑_{m=0}^{∞} \frac{(a_{1})_{m} ⋯ (a_{p})_{m}}{m! (b_{1})_{m} ⋯ (b_{q})_{m}} z^{m}

where (a_{1})_{m}, etc. are again Pochhammer symbols, Equation (1) becomes

J_{k} = \frac{(N + 1)_{k−1} (a + N − 1)_{k}}{k! (2N + a + b − 2)_{k}} _{4}F_{3}[ \begin{matrix}1 − N, 1 − k, 2 − a − N, 3 − a − b − k − 2N \\ 2 − a − k − N, 1 − k − N, 3 − a − b − 2N \end{matrix} ; 1] . (2)

The point of this conversion is that one may now look in the literature of hypergeometric series, which provides a large arsenal of utilities, such as transformation formulae for hypergeometric series etc., and see whether one (or a combination of more) of these formulae enable us to convert (2) into a different form from which the asymptotics of J_{k} can be directly read off. As I show below, this is indeed the case. To the informed reader it will come as no surprise that the computations below have been carried out using the Mathematica package HYP [5].

The crucial observation to make is that the _{4}F_{3}-series in (2) is not just an “arbitrary” hypergeometric series, but that it is a (−1)-balanced series (cf. [1, p. 140]), that is, that the sum of the lower parameters of the hypergeometric series minus the sum of the upper parameters is −1. Unfortunately, there are no transformation formulae available for (−1)- balanced series, but there are plenty of them available for 1-balanced series (or, as is common to say, for balanced series), such as, for example (see [4, Appendix (III.16), q ↑ 1]; this is T4302 in HYP),

4F_{3}[ \begin{matrix}A, B, C, −n \\ E, F, 1 + A + B + C − E − F − n \end{matrix}; 1]

= \frac{(A)_{n} (E + F − A − B)_{n} (E + F − A − C)_{n}}{(E)_{n} (F)_{n} (E + F − A − B − C)_{n}}

×_{4}F_{3}[ \begin{matrix}−n, E − A, F − A, E + F − A − B − C \\ E + F − A − B, E + F − A − C, 1 − A − n \end{matrix}; 1] , (3)

where n is a non-negative integer. That is, we are just by “2 off” in applying a transformation formula.

In order to relate series with (small) integral differences between the parameters, the hypergeometric literature offers contiguous relations. One of the simplest such relations is (this is C14 in HYP),

4F_{3}[ \begin{matrix}A, B, C, D \\ E, F, G \end{matrix}; z] = z \frac{BCD}{EF G}_{4}F_{3}[ \begin{matrix}A, B + 1, C + 1, D + 1 \\ E + 1, F + 1, G + 1 \end{matrix} ; z] +_{4}F_{3}[ \begin{matrix}A − 1, B, C, D \\ E, F, G \end{matrix}; z] .

If we iterate this contiguous relation, then we arrive at

4F_{3}[ \begin{matrix}A, B, C, D \\ E, F, G \end{matrix}; z] = z^{r} \frac{(B)_{r} (C)_{r} (D)_{r}}{(E)_{r} (F)_{r} (G)_{r}} _{4}F_{3}[ \begin{matrix}A, B + r, C + r, D + r \\ E + r, F + r, G + r \end{matrix}; z]

+ ∑_{s=0}^{r−1} z^{s} \frac{(B)_{s} (C)_{s} (D)_{s}}{(E)_{s} (F)_{s} (G)_{s}}_{4}F_{3}[ \begin{matrix}A − 1, B + s, C + s, D + s \\ E + s, F + s, G + s \end{matrix}; z] . (4)

We now apply (4) with r = k − 1 to the _{4}F_{3}-series in (2). The reader should observe that thereby all the _{4}F_{3}-series in the sum over s will be 0-balanced, while the single _{4}F_{3}-series outside of the sum will reduce to 1 because one of the upper parameters is 1 − k + r = 0. To be precise, after writing the result in compact form, application of (4) to (2) yields

J_{k} = ∑_{s=0}^{k−1} \frac{(N + 1)_{k−1} (a + N − 1)_{k} (1 − k)_{s} (2 − a − N)_{s} (3 − a − b − k − 2N)_{s}}{k! (a + b + 2N − 2)_{k} (2 − a − k − N)_{s} (1 − k − N)_{s} (3 − a − b − 2N)_{s}}

×_{4}F_{3}[ \begin{matrix}−N, 1 − k + s, 2 − a − N + s, 3 − a − b − k − 2N + s \\ 2 − a − k − N + s, 1 − k − N + s, 3 − a − b − 2N + s \end{matrix}; 1]

Continuing in the same spirit, if we apply the contiguous relation (4) a second time, this time with r = k − s − 1, then this will produce a double sum in which all appearing 4F_{3}-series are balanced. To be precise, we obtain

J_{k}

= ∑_{s=0}^{k−1} ∑_{t=0}^{k−s−1} \frac{(N + 1)_{k−1} (a + N − 1)_{k} (1 − k)_{s+t} (2 − a − N)_{s+t} (3 − a − b − k − 2N)_{s+t}}{k! (a + b + 2N − 2)_{k} (2 − a − k − N)_{s+t} (1 − k − N)_{s+t} (3 − a − b − 2N)_{s+t}}

×_{4}F_{3}[ \begin{matrix}3 − a − b − k − 2N + s + t, −1 − N, 2 − a − N + s + t, 1 − k + s + t \\ 3 − a − b − 2N + s + t, 1 − k − N + s + t, 2 − a − k − N + s + t \end{matrix}; 1] .

Now the transformation formula (3) can be applied, and, after some simplification, the resulting expression collapses to

J_{k} = ∑_{s=0}^{k−1} ∑_{t=0}^{k−s−1} \frac{(a − 1)_{k−s−t−1} (1 − a − N)_{s+t+1} (k − s − t)_{s+t} (s + t + 2)_{k−s−t−1}}{k! (2 − a − b − 2N)_{k}}

·_{4}F_{3}[ \begin{matrix}1 − k + s + t, k, a + b + N − 2, a + N \\ s + t + 2, a − 1, a + b + 2N − 1 \end{matrix} ; 1]

= ∑_{m=0}^{k−1} \frac{(a − 1)_{k−m−1} (1 − a − N)_{m+1} (k − m)_{m} (m + 1)_{k−m}}{k! (2 − a − b − 2N)_{k}}

·_{4}F_{3}[ \begin{matrix}−k + m + 1, k, a + b + N − 2, a + N \\ m + 2, a − 1, a + b + 2N − 1 \end{matrix} ; 1] .

Now the limit N, a, b → ∞, in which a ∼ a_{1}N and b ∼ b_{1}N, can be safely performed. Hence, we have proven the following result.

Theorem. The limit of the quantity J_{k} as N, a, b → ∞ such that a ∼ a_{1}N and b ∼ b_{1}N is equal to

lim_{N→∞} J_{k} = ∑_{m=0}^{k−1}(−1)^{k−m−1}\binom{k − 1}{m} (\frac{a_{1}}{a_{1} + b_{1} + 2})^{k} (\frac{a_{1} + 1}{a_{1}})^{m+1}

· ∑_{ℓ=0}^{k−m−1}(−1)^{ℓ}\binom{k − m − 1}{ℓ}\frac{(k + ℓ − 1)!(m + 1)!}{(k − 1)!(m + ℓ + 1)!} (\frac{(a_{1} + 1)(a_{1} + b_{1} + 1)}{a_{1}(a_{1} + b_{1} + 2)})^{ℓ} . (5)

This theorem does not give the same expression as in [2, Theorem IV.4]. However, once one has found some expression, it is easy to pass to an equivalent one by using hypergeometric transformation and summation formulae. (Since, in the above derivation, there is plenty of room for variation — for example, the transformation formula (3) could be applied in different ways, or we could have applied a different transformation formula for balanced _{4}F_{3}-series, see [6, Sec. 4.3.5] —, it is conceivable that there is a direct route leading to the corollary below, using the above ideas. We did not pursue this, however; in particular, we did not do a systematic search of such an alternative route.)

Corollary ([2, Theorem IV.4]). The limit of the quantity J_{k} as N, a, b → ∞ such that a ∼ a_{1}N and b ∼ b_{1}N is equal to

lim_{N→∞} J_{k} = \frac{1}{k} ∑_{j=0}^{k−1}(−1)^{j}\binom{k + j − 1}{j}\frac{(a_{1} + 1)^{j+1}}{(a_{1} + b_{1} + 2)^{k+j}} ∑_{i=0}^{k−j−1} \binom{k}{i}\binom{k}{i + j + 1}(a_{1} + 1)^{i}.

Proof. In the expression on the right-hand side of (5) we interchange the summations over m and ℓ, and we write the (now) inner sum over m in hypergeometric notation. This leads to

lim_{N→∞} J_{k} = ∑_{ℓ=0}^{k−1}(−1)^{k−ℓ−1} \frac{a_{1}^{k−ℓ−1} (a_{1} + 1)^{ℓ+1} (a_{1} + b_{1} + 1)^{ℓ}}{(a_{1} + b_{1} + 2)^{k+ℓ}} \frac{(k + ℓ − 1)!}{ℓ! (ℓ + 1)! (k − ℓ − 1)!}

·_{2}F_{1}[ \begin{matrix}2, −k + ℓ + 1 \\ ℓ + 2 \end{matrix} ; \frac{a_{1} + 1}{a_{1}}] .

To the _{2}F_{1}-series we apply the transformation formula (see [6, (1.7.1.3), sum reversed at the right-hand side]; this is T2106 in HYP)

2F_{1}[ \begin{matrix}A, −n \\ C \end{matrix}; z] = z^{n} \frac{(C − A)_{n}}{(C)_{n}}_{2}F_{1}[ \begin{matrix}−n, 1 − C − n \\ 1 + A − C − n \end{matrix}; −\frac{1 − z}{z} ] ,

where n is a non-negative integer. Writing the resulting _{2}F_{1}-series as a sum over i, this gives

lim_{N→∞} J_{k} = ∑_{ℓ=0}^{k−1} ∑_{i=0}^{k−ℓ−1}(−1)^{k−ℓ−i−1}\binom{k − ℓ − 1}{i}\frac{(k + ℓ − 1)!}{(ℓ − 1)! ℓ! (k − ℓ − 1)! (k − i − 1)(k − i)}

· \frac{(a_{1} + 1)^{k−i} (a_{1} + b_{1} + 1)^{ℓ}}{(a_{1} + b_{1} + 2)^{k+ℓ}} .

Now we replace the term (a_{1} +b_{1} +1)^{ℓ} by ∑_{j=0}^{ℓ} \binom{ℓ}{j}(−1)^{j}(a_{1} +b_{1} +2)^{ℓ−j}, we make the sum over ℓ the inner-most sum, and finally write the sum over ℓ in hypergeometric notation. In this way, the last equation becomes

lim_{N→∞} J_{k} = ∑_{j=0}^{k−1} ∑_{i=0}^{k−j−1}(−1)^{k−i−1}\binom{k − j − 1}{i}\frac{(k + j − 1)!}{(j − 1)! j! (k − j − 1)! (k − i − 1)(k − i)}

· \frac{(a_{1} + 1)^{k−i}}{(a_{1} + b_{1} + 2)^{k+j}} _{2}F_{1}[ \begin{matrix}−k + i + j + 1, k + j \\ j \end{matrix}; 1] .

The _{2}F_{1}-series can be evaluated by the Chu-Vandermonde summation (see [6, (1.7.7); Appendix (III.4)]; this is S2101 in HYP)

2F_{1}[ \begin{matrix}A, −n \\ C \end{matrix}; 1] = \frac{(C − A)_{n}}{(C)_{n}} ,

where n is a non-negative integer. After finally replacing i by k − i − j − 1, the claimed result follows immediately upon minor rearrangement of terms. □

References

1. G. E. Andrews, R. A. Askey and R. Roy, Special functions, The Encyclopedia of Mathematics and Its Applications, vol. 71, Cambridge University Press, Cambridge, 1999.

2. C. Carré, M. Deneufchâtel, J.-G. Luque and P. Vivo, Asymptotics of Selberg-type integrals: The unitary case and Newton’s interpolation formula, preprint; arχiv:1003.5996.

3. P. J. Forrester and S. O. Warnaar, The importance of the Selberg integral, Bull. Amer. Math. Soc. (N.S.) 45 (2008), 489–534.

4. G. Gasper and M. Rahman, Basic hypergeometric series, Encyclopedia of Mathematics And Its Applications 35, Cambridge University Press, Cambridge, 1990.

5. HYP and HYPQ — Mathematica packages for the manipulation of binomial sums and hypergeometric series, respectively q-binomial sums and basic hypergeometric series, J. Symbol. Comput. 20 (1995), 737–744, the packages are freely available at http://www.mat.univie.ac.at/~kratt.

6. L. J. Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966.

7. E. T. Whittaker and G. N. Watson, A course of modern analysis, fourth edition, Cambridge University Press, Cambridge, 1927, reprinted by Cambdridge University Press, 1996.

Fakultät für Mathematik, Universität Wien, Nordbergstrasze 15, A-1090 Vienna, Austria.

WWW: http://www.mat.univie.ac.at/~kratt