Probabilistic derivation of a bilinear summation formula for the Meixner-Pollaczek polynominals
Using the technique of canonical expansion in probability theory, a bilinear summation formula is derived for the special case of the Meixner-Pollaczek polynomials {λn(k)(x)} which are defined by the generating function ∑n=0∞λn(k)(x)zn/n!=(1+z)12(x−k)/(1−z)12(x+k), |z|<1. These polynomials...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Hindawi Publishing Corporation
1980
|
| Subjects: | |
| Online Access: | http://eprints.um.edu.my/17452/1/LeePA_%281980%29.pdf |
| Summary: | Using the technique of canonical expansion in probability theory, a bilinear summation formula is derived for the special case of the Meixner-Pollaczek polynomials {λn(k)(x)} which are defined by the generating function
∑n=0∞λn(k)(x)zn/n!=(1+z)12(x−k)/(1−z)12(x+k), |z|<1.
These polynomials satisfy the orthogonality condition
∫−∞∞pk(x)λm(k)(ix)λn(k)(ix)dx=(−1)nn!(k)nδm,n, i=−1
with respect to the weight function
p1(x)=sech πx
pk(x)=∫−∞∞…∫−∞∞sech πx1sech πx2 … sech π(x−x1−…−xk−1)dx1dx2…dxk−1, k=2,3,… |
|---|