Using the technique of canonical expansion in probability theory, a bilinear summation formula is derived for the special case of the Meixner-Pollaczek polynomials { λ n ( k ) ( x ) } which are defined by the generating function ∑ n = 0 ∞ λ n ( k ) ( x ) z n / n ! = ( 1 + z ) 1 2 ( x − k ) / ( 1 − z ) 1 2 ( x + k ) ,       | z | < 1.

These polynomials satisfy the orthogonality condition ∫ − ∞ ∞ p k ( x ) λ m ( k ) ( i x ) λ n ( k ) ( i x ) d x = ( − 1 ) n n ! ( k ) n δ m , n ,       i = − 1 with respect to the weight function p 1 ( x ) = sech   π x p k ( x ) = ∫ − ∞ ∞ … ∫ − ∞ ∞ sech   π x 1 sech   π x 2   …   sech   π ( x − x 1 − … − x k − 1 ) d x 1 d x 2 … d x k − 1 ,       k = 2 , 3 , …