Bessel Polynomials by Emil Grosswald (auth.)

By Emil Grosswald (auth.)

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It can also be Indeed, now we obtain instead of (I): ~ (yn,p) (_b)k+l = n [ v n (n+v+a-2) . . (n+a-l) (-i) (v) (k+v+a-l) (a+l)a v=0 "'" and the sun, again with a-I = x, equals n (x÷l)... (x÷n-l) v=0 ( - )1 This sun vanishes for k < n. v n (v) ( n + v + x - l ) ( n + v + x - 2 ) . . (k+v+x+l). Indeed, let F(y) = n . 1"v'n" n+v+x-i [- ) [vjy ; then v=0 ~n-k=l F n 8yn-k--------~= v=0[ (-l)V(~)(n+v+x-l)'''(k+v+x+l)yk+V+X" The sun to be evaluated is the last one for y = i. F(y) = yn+x-i n~ On the other hand, n v = yn+x-l(l-y)n.

N + a - 1 ) r ( a ) (r-k-1):(n-r+k+D~r-k-lr(a+r-1) n! (n+v+a-2) . . (n+a-1) • ( n - v + l ) . . (n+k+l) v! a ( a + l ) . . ( a + v + k - 1 ) n nl ~ ( ' l ) V v! v=0 (-b) k+l (n+k+l) ! (n+v+a-2) . . (n+a-1) (n-v) ' (k+v+a-1) (a+l)a . . (n+a-1) (k+v+a-1) (a+l)a so that (1) Mk(Yn,p ) = (_b)k+l (k+n+a-1) (a+l)a "'" n 1" v ' n " ... (k+n+a-1). (k-n+l). THEOREM i. g. [68], p. 113); however, a proof of the claim follows. is based on two lemmas, the first of which is indeed well-known. LEMMA i. 4) in [ZO]).

Function Jk+l/2(z) and t h e BP yn(~ i / z ) from the d i f f e r e n t i a l = The r e l a t i o n follows between t h e Bessel (see Chapters 2 and 3), e . g . , Kn+l/2(z-1. ) = (nz/2) 1/2e-1/Zyn(Z ) and t h e r e l a t i o n s in [68], d i r e c t l y (z-l) (see [68], (ii)) one by between K (z) and J ( z ) , or, as equations. 7. We formalize some of the results obtained so far in this chapter in the following theorem. THEOREM i. (-z/b)nh (-2n-a+l)(b/z) n (vi) yn(Z) = n~(-z/2)nL (-2n-l)(2/z). n (vii) Yn(Z;a,b) = eb/2Z(z/b)l-a/2w (viii) yn(Z) = e I/z W (ix) 8n(Z;a,b) l-a/2, ~1 (a-l)+n (b/z); 1 (2/z); O,n + ~- = (-1)nn'b-nL (-2n-a+l)(bz)" n (x) Sn(Z ) = ( - 1 ) n n !

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