Bessel Polynomials by Emil Grosswald (auth.)

By Emil Grosswald (auth.)

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We easily find that it has a single solution regular at z = 0 and normalized by y(0) = i, given by (26) Y(Z)(=Y-l(z;a'b)) = m=0~ Cm zm, with cm = (-l) m This solution reduces to a polynomial if and only if excess of 2. 12). The other case of interest is a = I, when y_l(Z;l,b) = l+z/b = yl(Z;l,b) and, by (20) of exact degree n. with a = i, y_n(Z;l,b) is a polynomial However, the theory for a = 1 has never been developed much further, the suggestion of Krall and Frink ([68], bottom of page 109) notwithstanding.

The formulae f o r yn(Z) and On(Z ) a r e i n d i c a t e d , because o f t h e i r r a t h e r f r e q u e n t use. 39 8. In the preceding sections we have discussed the relations of BP to other special functions essentially from the point of view of identifying BP with these other functions, for particular values of their parameters. A different kind of connection is pointed out by the following remark that, apparently, was never made before. Let Pn(X) be the n - t h Legendre Pol>momial. case of Jacobi Polynomials, (7) This i s , of course, a p a r t i c u l a r but can also be defined by the generating function (I-2zx+z2) -I/2 = n ~ Pn(X)Z .

See [17]). i 2~i I[zl=l Yk(Z;a'b)Yn (z;a'b)0(z;a'b)dz = 0. if k ~ n, then Let k < n; then yk{z;a,b)__ = k [ flkjzm,: ~ with 0 < m < n; m m=0 hence, the result follows from Corollary I. COROLLARY 6. (see [68], [17]). 1 2~i Proof. ~ bn! 28). By setting a = b = 2, one obtains from Corollary 6 also COROLLARY 7. 3. [see [68]) 1 2~1 2 IIzI=l Yn (z)e-2/zdz = (-l)n+l 2 2n+l " - - If we replace yn(Z;a,b) by zn@n(z-l;a,b) in Theorem i and its cdrollaries, we obtain the following results. 3~ THEOREM 2. r(a) r(a+r-1) Pl ( z ; a ' b ) = z-2 ~ r;0 Let (-bz) r a n d s e t = Mk(On(Z;a'b);Pl ( z ; a ' b ) ) = 2 -1~ f l z [ = l %(en;01) z -k (z-nen ( z ; a , b ) ) Ol(z;a,b)dz; then ~lk(On;O1) = (_b)k+l COROLLARY 8.

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