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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.11 Relations to Other Functions

Contents

§18.11(i) Explicit Formulas

See §§18.5(i) and 18.5(iii) for relations to trigonometric functions, the hypergeometric function, and generalized hypergeometric functions.

Ultraspherical

18.11.1 Pnm(x)=(12)m(-2)m(1-x2)12mCn-m(m+12)(x)=(n+1)m(-2)-m(1-x2)12mPn-m(m,m)(x),
0mn.

For the Ferrers function Pnm(x), see §14.3(i).

Compare also (14.3.21) and (14.3.22).

Laguerre

18.11.2 Ln(α)(x)=(α+1)nn!M(-n,α+1,x)=(-1)nn!U(-n,α+1,x)=(α+1)nn!x-12(α+1)e12xMn+12(α+1),12α(x)=(-1)nn!x-12(α+1)e12xWn+12(α+1),12α(x).

For the confluent hypergeometric functions M(a,b,x) and U(a,b,x), see §13.2(i), and for the Whittaker functions Mκ,μ(x) and Wκ,μ(x) see §13.14(i).

Hermite

18.11.3 Hn(x) =2nU(-12n,12,x2)
=2nxU(-12n+12,32,x2)
=212ne12x2U(-n-12,212x).
18.11.4 Hen(x) =212nU(-12n,12,12x2)
=212(n-1)xU(-12n+12,32,12x2)
=e14x2U(-n-12,x).

For the parabolic cylinder function U(a,z), see §12.2.

§18.11(ii) Formulas of Mehler–Heine Type

Jacobi

18.11.5 limn1nαPn(α,β)(1-z22n2)=limn1nαPn(α,β)(coszn)=2αzαJα(z).

Laguerre

18.11.6 limn1nαLn(α)(zn)=1z12αJα(2z12).

Hermite

18.11.7 limn(-1)nn1222nn!H2n(z2n12) =1π12cosz,
18.11.8 limn(-1)n22nn!H2n+1(z2n12) =2π12sinz.

For the Bessel function Jν(z), see §10.2(ii). The limits (18.11.5)–(18.11.8) hold uniformly for z in any bounded subset of .