Selberg’s second beta integral and an integral of Mehta. (English) Zbl 0683.33001
Probability, statistics, and mathematics, Pap. in Honor of Samuel Karlin, 27-39 (1989).
[For the entire collection see Zbl 0679.00013.]
Selberg’s two beta integrals are \[ \int_{C_ n}| d_ n(t)|^ c w(t)dt, \] where \(D_ n(t)=\prod_{1\leq i<j\leq n}| t_ i-t_ j|^ 2\) is the discriminant, and \[ w(t)=\prod^{n}_{i=1}t_ i^{a-1}(1-t_ i)^{b-1},\quad C_ n=\{t_ i:\quad 0\leq t_ i\leq 1\} \] for the first integral and \[ w(t)=(1-\sum^{n}_{i=1}t_ i)^{b-1}\prod^{n}_{i=1}t_ i^{a- 1},\quad C_ n=\{t_ i:\quad t_ i\geq 0,\quad \sum^{n}_{i=1}t_ i\leq 1\} \] for the second. The second is evaluated here, using Aomoto’s idea to introduce a new parameter to first evaluate Selberg’s gamma integral, and then obtain the second beta integral from this as Jacobi evaluated Euler’s beta integral from the gamma integral. Integrals over groups can often be reduced to integrals over \(R^ n\) when the functions being integrated have appropriate symmetry. An extension of such a reduction found by Mehta for specific functions is given for products of two complex-valued functions f with \(f(S)=f(gSg^{-1})\), S a Hermitian \(n\times n\) matrix and \(g\in U(n)\).
Selberg’s two beta integrals are \[ \int_{C_ n}| d_ n(t)|^ c w(t)dt, \] where \(D_ n(t)=\prod_{1\leq i<j\leq n}| t_ i-t_ j|^ 2\) is the discriminant, and \[ w(t)=\prod^{n}_{i=1}t_ i^{a-1}(1-t_ i)^{b-1},\quad C_ n=\{t_ i:\quad 0\leq t_ i\leq 1\} \] for the first integral and \[ w(t)=(1-\sum^{n}_{i=1}t_ i)^{b-1}\prod^{n}_{i=1}t_ i^{a- 1},\quad C_ n=\{t_ i:\quad t_ i\geq 0,\quad \sum^{n}_{i=1}t_ i\leq 1\} \] for the second. The second is evaluated here, using Aomoto’s idea to introduce a new parameter to first evaluate Selberg’s gamma integral, and then obtain the second beta integral from this as Jacobi evaluated Euler’s beta integral from the gamma integral. Integrals over groups can often be reduced to integrals over \(R^ n\) when the functions being integrated have appropriate symmetry. An extension of such a reduction found by Mehta for specific functions is given for products of two complex-valued functions f with \(f(S)=f(gSg^{-1})\), S a Hermitian \(n\times n\) matrix and \(g\in U(n)\).
Reviewer: R.Askey
MSC:
33B15 | Gamma, beta and polygamma functions |
33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |
33C60 | Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions) |