18.17. ORTHOGONAL POLYNOMIALS 983 18.17.1-2. Generalized Laguerre polynomials. The generalized Laguerre polynomials L α n = L α n (x)(α >–1) satisfy the equation xy xx +(α + 1 – x)y x + ny = 0 and are defined by the formulas L α n (x)= 1 n! x –α e x d n dx n x n+α e –x = n m=0 C n–m n+α (–x) m m! = n m=0 Γ(n + α + 1) Γ(m + α + 1) (–x) m m!(n – m)! . Notation: L 0 n (x)=L n (x). Special cases: L α 0 (x)=1, L α 1 (x)=α + 1 – x, L –n n (x)=(–1) n x n n! . To calculate L α n (x)forn ≥ 2, one can use the recurrence formulas L α n+1 (x)= 1 n + 1 (2n + α + 1 – x)L α n (x)–(n + α)L α n–1 (x) . Other recurrence formulas: L α n (x)=L α n–1 (x)+L α–1 n (x), d dx L α n (x)=–L α+1 n–1 (x), x d dx L α n (x)=nL α n (x)–(n+α)L α n–1 (x). The functions L α n (x) form an orthogonal system on the interval 0 < x < ∞ with weight x α e –x : ∞ 0 x α e –x L α n (x)L α m (x) dx = 0 if n ≠ m, Γ(α+n+1) n! if n = m. The generating function is (1 – s) –α–1 exp – sx 1 – s = ∞ n=0 L α n (x)s n , |s| < 1. 18.17.2. Chebyshev Polynomials and Functions 18.17.2-1. Chebyshev polynomials of the first kind. The Chebyshev polynomials of the first kind T n = T n (x) satisfy the second-order linear ordinary differential equation (1 – x 2 )y xx – xy x + n 2 y = 0 (18.17.2.1) and are defined by the formulas T n (x)=cos(n arccos x)= (–2) n n! (2n)! √ 1 – x 2 d n dx n (1 – x 2 ) n– 1 2 = n 2 [n/2] m=0 (–1) m (n – m – 1)! m!(n – 2m)! (2x) n–2m (n = 0, 1, 2, ), where [A] stands for the integer part of a number A. 984 SPECIAL FUNCTIONS AND THEIR PROPERTIES An alternative representation of the Chebyshev polynomials: T n (x)= (–1) n (2n – 1)!! (1 – x 2 ) 1/2 d n dx n (1 – x 2 ) n–1/2 . The first five Chebyshev polynomials of the first kind are T 0 (x)=1, T 1 (x)=x, T 2 (x)=2x 2 – 1, T 3 (x)=4x 3 – 3x, T 4 (x)=8x 4 – 8x 2 + 1. The recurrence formulas: T n+1 (x)=2xT n (x)–T n–1 (x), n ≥ 2. The functions T n (x) form an orthogonal system on the interval –1 < x < 1, with 1 –1 T n (x)T m (x) √ 1 – x 2 dx = 0 if n ≠ m, 1 2 π if n = m ≠ 0, π if n = m = 0. The generating function is 1 – sx 1 – 2sx + s 2 = ∞ n=0 T n (x)s n (|s| < 1). The functions T n (x) have only real simple zeros, all lying on the interval –1 < x < 1. The normalized Chebyshev polynomials of the first kind, 2 1–n T n (x), deviate from zero least of all. This means that among all polynomials of degree n with the leading coefficient 1, it is the maximum of the modulus max –1≤x≤1 |2 1–n T n (x)| that has the least value, the maximum being equal to 2 1–n . 18.17.2-2. Chebyshev polynomials of the second kind. The Chebyshev polynomials of the second kind U n = U n (x) satisfy the second-order linear ordinary differential equation (1 – x 2 )y xx – 3xy x + n(n + 2)y = 0 and are defined by the formulas U n (x)= sin[(n + 1) arccos x] √ 1 – x 2 = 2 n (n + 1)! (2n + 1)! 1 √ 1 – x 2 d n dx n (1 – x 2 ) n+1/2 = [n/2] m=0 (–1) m (n – m)! m!(n – 2m)! (2x) n–2m (n = 0, 1, 2, ). The first five Chebyshev polynomials of the second kind are U 0 (x)=1, U 1 (x)=2x, U 2 (x)=4x 2 –1, U 3 (x)=8x 3 –4x, U 4 (x)=16x 4 –12x 2 +1. The recurrence formulas: U n+1 (x)=2xU n (x)–U n–1 (x), n ≥ 2. The generating function is 1 1 – 2sx + s 2 = ∞ n=0 U n (x)s n (|s| < 1). The Chebyshev polynomials of the first and second kind are related by U n (x)= 1 n + 1 d dx T n+1 (x). 18.17. ORTHOGONAL POLYNOMIALS 985 18.17.2-3. Chebyshev functions of the second kind. The Chebyshev functions of the second kind, U 0 (x) = arcsin x, U n (x)=sin(n arccos x)= √ 1 – x 2 n dT n (x) dx (n = 1, 2, ), just as the Chebyshev polynomials, also satisfy the differential equation (18.17.2.1). The first five the Chebyshev functions are U 0 (x)=0, U 1 (x)= √ 1 – x 2 , U 2 (x)=2x √ 1 – x 2 , U 3 (x)=(4x 2 – 1) √ 1 – x 2 , U 5 (x)=(8x 3 – 4x) √ 1 – x 2 . The recurrence formulas: U n+1 (x)=2x U n (x)–U n–1 (x), n ≥ 2. The functions U n (x) form an orthogonal system on the interval –1 < x < 1, with 1 –1 U n (x) U m (x) √ 1 – x 2 dx = 0 if n ≠ m or n = m = 0, 1 2 π if n = m ≠ 0. The generating function is √ 1 – x 2 1 – 2sx + s 2 = ∞ n=0 U n+1 (x)s n (|s| < 1). 18.17.3. Hermite Polynomials 18.17.3-1. Various representations of the Hermite polynomials. The Hermite polynomials H n = H n (x) satisfy the second-order linear ordinary differential equation y xx – 2xy x + 2ny = 0 and is defined by the formulas H n (x)=(–1) n exp x 2 d n dx n exp –x 2 = [n/2] m=0 (–1) m n! m!(n – 2m)! (2x) n–2m . The first five polynomials are H 0 (x)=1, H 1 (x)=2x, H 2 (x)=4x 2 –2, H 3 (x)=8x 3 –12x, H 4 (x)=16x 4 –48x 2 +12. Recurrence formulas: H n+1 (x)=2xH n (x)–2nH n–1 (x), n ≥ 2; d dx H n (x)=2nH n–1 (x). Integral representation: H 2n (x)= (–1) n 2 2n+1 √ π exp x 2 ∞ 0 exp –t 2 t 2n cos(2xt) dt, H 2n+1 (x)= (–1) n 2 2n+2 √ π exp x 2 ∞ 0 exp –t 2 t 2n+1 sin(2xt) dt, where n = 0, 1, 2, 986 SPECIAL FUNCTIONS AND THEIR PROPERTIES 18.17.3-2. Orthogonality. The generating function. An asymptotic formula. The functions H n (x) form an orthogonal system onthe interval –∞<x <∞with weight e –x 2 : ∞ –∞ exp –x 2 H n (x)H m (x) dx = 0 if n ≠ m, √ π 2 n n!ifn = m. Generating function: exp –s 2 + 2sx = ∞ n=0 H n (x) s n n! . Asymptotic formula as n →∞: H n (x) ≈ 2 n+1 2 n n 2 e – n 2 exp x 2 cos √ 2n + 1 x – 1 2 πn . 18.17.3-3. Hermite functions. The Hermite functions h n (x) are introduced by the formula h n (x)=exp – 1 2 x 2 H n (x)=(–1) n exp 1 2 x 2 d n dx n exp –x 2 , n = 0, 1, 2, The Hermite functions satisfy the second-order linear ordinary differential equation h xx +(2n + 1 – x 2 )h = 0. The functions h n (x) form an orthogonal system on the interval –∞ < x < ∞ with weight 1: ∞ –∞ h n (x)h m (x) dx = 0 if n ≠ m, √ π 2 n n!ifn = m. 18.17.4. Jacobi Polynomials and Gegenbauer Polynomials 18.17.4-1. Jacobi polynomials. The Jacobi polynomials, P α,β n (x), are solutions of the second-order linear ordinary differ- ential equation (1 – x 2 )y xx + β – α –(α + β + 2)x y x + n(n + α + β + 1)y = 0 and are defined by the formulas P α,β n (x)= (–1) n 2 n n! (1 – x) –α (1 + x) –β d n dx n (1 – x) α+n (1 + x) β+n = 2 –n n m=0 C m n+α C n–m n+β (x – 1) n–m (x + 1) m , where the C a b are binomial coefficients. 18.17. ORTHOGONAL POLYNOMIALS 987 The generating function: 2 α+β R –1 (1 – s + R) –α (1 + s + R) –β = ∞ n=0 P α,β n (x)s n , R = √ 1 – 2xs + s 2 , |s| < 1. The Jacobi polynomials are orthogonal on the interval –1 ≤ x ≤ 1 with weight (1–x) α (1+x) β : 1 –1 (1 – x) α (1 + x) β P α,β n (x)P α,β m (x) dx = ⎧ ⎨ ⎩ 0 if n ≠ m, 2 α+β+1 α + β + 2n + 1 Γ(α + n + 1)Γ(β + n + 1) n! Γ(α + β + n + 1) if n = m. For α >–1 and β >–1, all zeros of the polynomial P α,β n (x) are simple and lie on the interval –1 < x < 1. 18.17.4-2. Gegenbauer polynomials. The Gegenbauer polynomials (also called ultraspherical polynomials), C (λ) n (x), are solu- tions of the second-order linear ordinary differential equation (1 – x 2 )y xx –(2λ + 1)xy x + n(n + 2λ)y = 0 and are defined by the formulas C (λ) n (x)= (–2) n n! Γ(n + λ) Γ(n + 2λ) Γ(λ) Γ(2n + 2λ) (1 – x 2 ) –λ+1/2 d n dx n (1 – x 2 ) n+λ–1/2 = [n/2] m=0 (–1) m Γ(n – m + λ) Γ(λ) m!(n – 2m)! (2x) n–2m . Recurrence formulas: C (λ) n+1 (x)= 2(n + λ) n + 1 xC (λ) n (x)– n + 2λ – 1 n + 1 C (λ) n–1 (x); C (λ) n (–x)=(–1) n C (λ) n (x), d dx C (λ) n (x)=2λC (λ+1) n–1 (x). The generating function: 1 (1 – 2xs + s 2 ) λ = ∞ n=0 C (λ) n (x)s n . The Gegenbauer polynomials are orthogonal on the interval –1 ≤ x ≤ 1 with weight (1 – x 2 ) λ–1/2 : 1 –1 (1 – x 2 ) λ–1/2 C (λ) n (x)C (λ) m (x) dx = ⎧ ⎨ ⎩ 0 if n ≠ m, πΓ(2λ + n) 2 2λ–1 (λ + n)n! Γ 2 (λ) if n = m. 988 SPECIAL FUNCTIONS AND THEIR PROPERTIES 18.18. Nonorthogonal Polynomials 18.18.1. Bernoulli Polynomials 18.18.1-1. Definition. Basic properties. The Bernoulli polynomials B n (x) are introduced by the formula B n (x)= n k=0 C k n B k x n–k (n = 0, 1, 2, ), where C k n are the binomial coefficients and B n are Bernoulli numbers (see Subsec- tion 18.1.3). The Bernoulli polynomials can be defined using the recurrence relation B 0 (x)=1, n–1 k=0 C k n B k (x)=nx n–1 , n = 2, 3, The first six Bernoulli polynomials are given by B 0 (x)=1, B 1 (x)=x – 1 2 , B 2 (x)=x 2 – x + 1 6 , B 3 (x)=x 3 – 3 2 x 2 + 1 2 x, B 4 (x)=x 4 – 2x 3 + x 2 – 1 30 , B 5 (x)=x 5 – 5 2 x 4 + 5 3 x 3 – 1 6 x. Basic properties: B n (x + 1)–B n (x)=nx n–1 , B n+1 (x)=(n + 1)B n (x), B n (1 – x)=(–1) n B n (x), (–1) n E n (–x)=E n (x)+nx n–1 , where the prime denotes a derivative with respect to x,andn = 0, 1, Multiplication and addition formulas: B n (mx)=m n–1 m–1 k=0 B n x + k m , B n (x + y)= n k=0 C k n B k (x)y n–k , where n = 0, 1, and m = 1, 2, 18.18.1-2. Generating function. Fourier series expansions. Integrals. The generating function is expressed as te xt e t – 1 ≡ ∞ n=0 B n (x) t n n! (|t| < 2π). This relation may be used as a definition of the Bernoulli polynomials. 18.18. NONORTHOGONAL POLYNOMIALS 989 Fourier series expansions: B n (x)=–2 n! (2π) n ∞ k=1 cos(2πkx – 1 2 πn) k n ,(n = 1, 0 < x < 1; n > 1, 0 ≤ x ≤ 1); B 2n–1 (x)=2(–1) n (2n – 1)! (2π) 2n–1 ∞ k=1 sin(2kπx) k 2n–1 (n = 1, 0 < x < 1; n > 1, 0 ≤ x ≤ 1); B 2n (x)=2(–1) n (2n)! (2π) 2n ∞ k=1 cos(2kπx) k 2n (n = 1, 2, , 0 ≤ x ≤ 1). Integrals: x a B n (t) dt = B n+1 (x)–B n+1 (a) n + 1 , 1 0 B m (t)B n (t) dt =(–1) n–1 m! n! (m + n)! B m+n , where m and n are positive integers and B n are Bernoulli numbers. 18.18.2. Euler Polynomials 18.18.2-1. Definition. Basic properties. Definition: E n (x)= n k=0 C k n E k 2 n x – 1 2 n–k (n = 0, 1, 2, ), where C k n are the binomial coefficients and E n are Euler numbers. The first six Euler polynomials are given by E 0 (x)=1, E 1 (x)=x – 1 2 , E 2 (x)=x 2 – x, E 3 (x)=x 3 – 3 2 x 2 + 1 4 , E 4 (x)=x 4 – 2x 3 + x, E 5 (x)=x 5 – 5 2 x 4 + 5 2 x 2 – 1 2 . Basic properties: E n (x + 1)+E n (x)=2x n , E n+1 =(n + 1)E n (x), E n (1 – x)=(–1) n E n (x), (–1) n+1 E n (–x)=E n (x)–2x n , where the prime denotes a derivative with respect to x,andn = 0, 1, Multiplication and addition formulas: E n (mx)=m n m–1 k=0 (–1) k E n x + k m , n = 0, 1, , m = 1, 3, ; E n (mx)=– 2 n + 1 m n m–1 k=0 (–1) k E n+1 x + k m , n = 0, 1, , m = 2, 4, ; E n (x + y)= n k=0 C k n E k (x)y n–k , n = 0, 1, . 2m)! (2x) n–2m (n = 0, 1, 2, ), where [A] stands for the integer part of a number A. 984 SPECIAL FUNCTIONS AND THEIR PROPERTIES An alternative representation of the Chebyshev polynomials: T n (x)= (–1) n (2n. Chebyshev polynomials of the first kind, 2 1–n T n (x), deviate from zero least of all. This means that among all polynomials of degree n with the leading coefficient 1, it is the maximum of the modulus. polynomials of the first and second kind are related by U n (x)= 1 n + 1 d dx T n+1 (x). 18.17. ORTHOGONAL POLYNOMIALS 985 18.17.2-3. Chebyshev functions of the second kind. The Chebyshev functions of