Recursive and Explicit Formulas for Expansion and Connection Coefficients in Series of Classical Orthogonal Polynomial Products

University of Pennsylvania, Perelman School of Medicine, Philadelphia, PA, USA 2 m 1 b m 1 P i ( b )= t m 2 Q j (t) = zm3Rk(z) = & sum; n 1 =0 2 m 2 & sum; n 2 =0 2 m 3 & sum; n 3 =0 a m 1 , n1(i)Pi+m1-n1(b), a m 2 , n2(j)Qj+m2-n2(t), a m 3 ,n 3 ( k ) R k + m 3 -n 3 ( z ) , we find the coefficients b ( p 1 , p 2 , m 1 , m 2 ) and b ( p 1 , p 2 , p 3 , m 1 , m 2 , m 3 ) i, j i, j, k in the two expansions = b m 1 t m 2 D p 1 b D t p 2 f ( b , t )= b m 1 t m 2 f ( p 1 , p 2 ) ( b , t ) = infinity & sum; i, j =0 b ( p 1 , p2, m1, m 2 ) i, j P i ( b ) Q j ( t ) , b m 1 t m 2 z m 3 D p 1 bDtp2Dp3 z f(b , t , z ) = b m 1 t m 2 z m 3 f ( p 1 ,p 2 , p 3 ) ( b , t , z ) infinity & sum; i, j =0 b ( p 1 , p2, p3, m1, m2, m 3 ) i, j, k P i ( b ) Q j ( t ) R k ( z ) . We apply these findings to reduce partial differential equations by converting polynomial coefficients to recursive formulas in the solution's expansion coefficients.