# SPECIAL FUNCTIONS and POLYNOMIALS

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1 SPECIAL FUNCTIONS and POLYNOMIALS Gerard t Hooft Stefan Nobbenhuis Institute for Theoretical Physics Utrecht University, Leuvenlaan CC Utrecht, the Netherlands and Spinoza Institute Postbox TD Utrecht, the Netherlands Many of the special functions and polynomials are constructed along standard procedures In this short survey we list the most essential ones. April 8, 13 1

2 1 Legendre Polynomials P l x). Differential Equation: 1 x ) P l x) x P l x) + ll + 1) P l x) =, or d 1 x ) d P lx) + ll + 1) P l x) =. 1.1) Generating function: Orthonormality: Expressions forp l x) : P l x)t l = 1 xt + t ) 1 for t < 1, x 1. 1.) l= 1 P l x) P l x) = -1 l + 1 δ l l, 1.3) P l x)p l x )l + 1) = δx x ). 1.4) l= Recurrence relations: P l x) = 1 [l/] 1) ν l ν)! l ν= ν! l ν)! l ν)! xl ν 1.5) = 1 d ) l x 1) l, 1.6) l! l = 1 π π x + x 1 cos ϕ) l dϕ. 1.7) l P l 1 l + 1) x P l + l + 1) P l+1 = ; P l = x P l 1 + x 1 P l 1 ; l xp l l P l = P l 1 ; xp l + l + 1) P l = P l+1 ; d [P l+1 P l 1 ] = l + 1) P l. 1.8) P = 1, P 1 = x, P = 1 3x 1), P 3 = 1 x5x 3). 1.9) 1

3 Associated Legendre Functions P m l x). 1 x ) Pl m x) x Pl m x) + ll + 1) m ) P m 1 x l x) =..1) Generating function: l=m 1-1 l l= m= P m l x) z m y l m! = [ 1 y x + z 1 x ) + y ] 1..) Pl m x) Pl m l + m)! x) = l + 1 l m)! δ l l, l, l m )..3) l m)! l + 1) l + m)! P l m x) Pl m x ) = δx x ), x < 1 and x < 1 )..4) Expressions for P m l x) 1 : P m l x) = Recurrence relations: ) m Pl m x) = 1 x ) 1 d m P l x)..5) l + m)! π 1) m/ x + x 1 cos ϕ ) l cos mϕ dϕ..6) l! π P m+1 l mx 1 x P m l + {ll + 1) mm 1)}P m 1 l =.7) 1 x P m+1 l x) = 1 x ) P m l x) + mx P m l x), l + 1)x P m l = l + m) P m l 1 + l + 1 m) P m l+1,.8) x P m l = P m l 1 l + 1 m) 1 x P m 1 l, and various others. Pl+1 m Pl 1 m = l + 1) Pl m 1 1 x,.9) P 1 1 = 1 x, P = 31 x ), P 1 = 3x 1 x, P 3 = 15 x1 x )..1) 1 Note that some authors define Pl mx) with a factor 1)m, giving Pl mx) = 1)m 1 x ) 1 m d m ) Pl x). Obviously this minus sign propagates to the generating function, the recurrence relations and the explicit examples, when m is odd.

4 3 Bessel J n x) and Hankel H n x) functions. Differential equation for both J n and H n ): Generating function if n integer): x J nx) + x J nx) + x n ) J n x) =. 3.1) n= J n αx) J n = 1) n J n. ) s n = e x α s ) s, 3.) α ξ J n αξ) J n βξ) dξ = 1 δ α β ). 3.3) α a ξ J n αξ) J n βξ) dξ = a {J n+1αa)} δ αβ. 3.4) if in the nd relation α, β are roots of the equation J n αξ) =. Expressions for J n x) for n integer): J n x) = J n x) = 1 π 1) k k! k + n)! k= π x ) n+k = Recurrence relations for both J n and H n ): 1 x ) n πi t n 1 dt e t x /4t. 3.5) cos nθ x sin θ) dθ. 3.6) d {xn J n x)} = x n J n 1 x) ; J n 1 x) + J n+1 x) = n x J nx) ; J nx) = J n 1 x) n x J nx) = n x J nx) J n+1 x) = 1 J n 1x) J n+1 x)). 3.7) Relations between Hankel and Bessel functions: H n 1) i x) = e nπi J n x) J n x) ) ; sin nπ H n ) i x) = e nπi J n x) J n x) ), 3.8) sin nπ so that J n x) = 1 H 1) n x) + H n ) x) ) ; J n x) = 1 e nπi H n 1) x) + e nπi H n ) x) ). 3.9) 3

5 4 Spherical Bessel Functions j l x). Generating Function: xj l ) + x ll + 1) ) jl =. 4.1) x j l x) t l l= l! = j x xt ). 4.) x j l αx) j l βx) = π δα β). 4.3) α Expressions for j l : j l x) j l x) = π j l x) = x J l+ 1 x) = 1) l x l π l + 1 δ ll. 4.4) d x ) l sin x x, 4.5) j l x) = = x l 1 l+1 l! 1 l l! l + 1)! xl e ixs 1 s ) l ds 1 1 x) 1 x) ) ! l + 3)!l + 3)l + 5). 4.6) Recurrence relations: j l+1 = l x j l j l = l + 1 j l j l ) x j x) = sin x x ; j 1 x) = sin x x j x) = 3 sin x x 3 cos x ; x 3 cos x x sin x x. 4.8) 4

6 5 Hermite Polynomials H n x). or d Generating function: More general: H nx) x H nx) + n H n x) =, 1 Hn x) e x) + n x + 1) H n x) e 1 x =. 5.1) H n x) s n /n! = e s +sx. 5.) n= H n x) H m x) e x = n n! π δ nm 5.3) H n x) H n y)/ n n!) = π δx y) e x. 5.4) n= H n x) H n y) s n / n n!) = n= Expressions for H n : 1 1 s exp s x + y ) + sxy 1 s ). 5.5) H n x) = 1) n H n x); 5.6) H n x) = 1) n e x d ) ne x = e 1 x x d ) ne 1 x ; 5.7) Recurrence relations: n/ H n x) = 1) n/ n! 1) k x) k k)! 1n k)!, H n x) = 1) n 1 k= n 1 n! k= d m H n x) m = if n even, 1) k x) k+1 k + 1)! n 1 k ), if n odd. 5.8)! m n! n m)! H n mx), 5.9) x H n x) = 1H n+1x) + n H n 1 x), 5.1) H n x) = x d ) H n 1 x). 5.11) H x) = 1, H 1 x) = x, H x) = 4x. 5.1) 5

7 6 Laguerre Polynomials L n x). Generating function: x L nx) + 1 x) L nx) + n L n x) =. 6.1) n= L n x) z n = 1 1 z e xz 1 z. 6.) Expressions for L n : L n x) L m x) e x = δ nm. 6.3) L n x) = ex d n! Recurrence relation: = 1)n n! ) n x n e x ) x n n 1! xn 1 + n n 1) x n ) n n! ). 6.4)! 1 + n x) L n n L n 1 n + 1)L n+1 = ; x L nx) = n L n x) n L n 1 x). 6.5) L x) = 1 ; L 1 x) = 1 x ; L x) = 1! x 4x + ). 6.6) It s important to note that sometimes different definitions are used for the Laguerre and Associated Laguerre polynomials, where the Generating Function has the form: n= L nx) z n /n! = 1 this case the Expressions given for L n should be multiplied by n!. 1 z e xz 1 z. In 6

8 7 Associated Laguerre Polynomials L k nx). Generating function: x L k n + k + 1 x) L k n + n L k n =. 7.1) L k nx) z n = n= 1 xz e 1 z. 7.) 1 z) k+1 L k nx) z n u k k= n=k k! = 1 1 z exp xz + u). 7.3) 1 z L k nx) L k mx) x k e x = n + k)! δ nm. 7.4) n! Expressions for L k n : L k nx) = 1) k d ) k Ln+k x). 7.5) Recurrence relation: L k nx) = ex x k n! d ) n x n+k e x ). 7.6) L k n 1 x) + Ln k 1 x) = L k n x) ; x L k n x) = n L k n x) n + k)l k n 1 x). 7.7) L k x) = 1 ; L k 1x) = x + k + 1 ; L k x) = 1 [ x k + )x + k + 1)k + ) ] ; L k 3x) = 1 [ x 3 + 3k + 3)x 3k + )k + 3)x + k + 1)k + )k + 3) ]. 7.8) 6 7

9 8 Tschebyscheff 3 Polynomials T n x). Generating function: Symmetry relation: 1 x ) d T nx) x d T nx) + n T n x) =. 8.1) T n x) y n = n= 1 xy 1 xy + y. 8.) T n x) = T n x). 8.3) Expression for T n : 1 1 { T m x)t n x) = 1 x 1 πδ nm m, n π m = n = 8.4) T n x) = cosn cos 1 x) 8.5) [{ } n { } n ] x + i 1 x + x i 1 x. 8.6) Recurrence relation: T n x) = 1 T n+1 x T n x) + T n 1 = 8.7) 1 x )T nx) = nx T n x) + n T n 1 x). 8.8) T x) = 1 ; T 1 x) = x ; T x) = x 1 ; T 3 x) = 4x 3 3x. 8.9) 3 Transliterations Chebyshev and Tchebicheff also occur. 8

10 9 Remark. All of the functions discussed here are special cases of hypergeometric functions mf n a 1, a,... a m ; b 1, b,... b n ; z) defined by: where Differential equations: m = n = 1: m =, n = 1: mf n a 1, a,... a m ; b 1, b,... b n ; z) = a) r r=p a 1 ) r a ) r... a m ) r z r b 1 ) r b ) r... b n ) r r!, 9.1) Γa + r) ; r a positive integer. 9.) Γa) z d ) 1F 1 + b z) d dz dz 1 F 1 a 1 F 1 =. 9.3) z1 z) d ) F 1 + c a + b + 1)z ) d dz dz F 1 ab F 1 =. 9.4) We have: P l x) = F 1 l, l + 1; 1; 1 x ) ; 9.5) Pl m l + m)! 1 x ) m/ x) = F l m)! m 1 m l, m + l + 1; m + 1; 1 x ) ; m! 9.6) ) J n x) = e ix x n 1F 1 n + n! 1; n + 1; ix) ; 9.7) H n x) = 1) n n)! 1F 1 n; n! 1; x ) ; n n + 1)! H n+1 x) = 1) x 1 F 1 n; n! 3; x ) ; L n x) = 1 F 1 n; 1; x) ; 9.8) 9.9) 9.1) L k Γn + k + 1) nx) = n!γk + 1) 1 F 1 n; k + 1; x) ; 9.11) T n x) = F 1 n, n; 1; 1 x ). 9.1) 9

11 References [1] H. Margenau and G.M. Murphy, The Mathematics of Physics and Chemistry, D. van Nostrand Comp. Inc., 1943, [] I.S. Gradshteyn and I.W. Ryzhik, Tables of Integrals, Series and Products, Acad. Press, New York, San Francisco, London, [3] W.W. Bell, Special Functions for Scientists and Engineers, D. van Nostrand Comp. Ltd.,

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