A conjunctive canonical expansion of multiple-valued functions

2003 ◽  
Author(s):  
E. Dubrova ◽  
P. Farm
Keyword(s):  
Canonical Expansion

scholarly journals Probabilistic derivation of a bilinear summation formula for the Meixner-Pollaczek polynominals

1980 ◽  
Vol 3 (4) ◽  
pp. 761-771 ◽  
Author(s):  
P. A. Lee
Keyword(s):  
Probability Theory ◽  
Weight Function ◽  
Generating Function ◽  
Summation Formula ◽  
Orthogonality Condition ◽  
Canonical Expansion ◽  
Pollaczek Polynomials ◽  
Special Case

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=−1with respect to the weight functionp1(x)=sech πxpk(x)=∫−∞∞…∫−∞∞sech πx1sech πx2 … sech π(x−x1−…−xk−1)dx1dx2…dxk−1,   k=2,3,…

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