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The multinomial distribution and some Bergman kernels. (English) Zbl 1102.41028
Chanillo, Sagun (ed.) et al., Geometric analysis of PDE and several complex variables. Dedicated to François Trèves. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3386-3/pbk). Contemporary Mathematics 368, 249-265 (2005).
The first section of this article derives an asymptotic expansion as \(k\to\infty\) of the binomial sum \(\sum_{j=0}^k \binom{k}{j} x^j (1-x)^{k-j} f(j/k)\), where \(f\in C^\infty([0,1])\), and of a corresponding integral. The second section gives analogous expansions in higher dimensions for multinomial sums and integrals. The application in the third section is to compute explicitly the singularities of the Bergman kernel function of the unit ball with respect to weight functions depending only on the moduli of the coordinates.
For the entire collection see [Zbl 1058.35003].

41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)