Handbook of mathematics for engineers and scienteists part 146 ppsx

, just as the Chebyshev polynomials, also satisfy the differential equation 18.17.2.1... The generating function... Fourier series expansions.

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: L0n (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) for n≥ 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– n 1(x), dx d L α n (x) = –L α+ n–11(x), x dx d 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 interval0< x < ∞ with weight

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–1exp



– 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– x2)y xx  – xy x  + n2y=0 (18.17.2.1) and are defined by the formulas

T n (x) = cos(n arccos x) = (–2)n n!

(2n)!

√

1– x2 d n

dx n



(1– x2)n–12

= 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.

An alternative representation of the Chebyshev polynomials:

T n (x) = (–1)n

(2n–1)!!(1– x2)1 2

d n

dx n(1– x2)n–1 2 The first five Chebyshev polynomials of the first kind are

T0(x) =1, T1(x) = x, T2(x) =2x2–1, T3(x) =4x3–3x, T4(x) =8x4–8x2+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– x2 dx=

0

if n≠m, 1

2π if n = m≠ 0,

π if n = m =0

The generating function is

1– sx

1–2sx + s2 =

∞



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,21 –n T n (x), deviate from zero least of all This means that among all polynomials of degree n with the leading coefficient1,

it is the maximum of the modulus max

– 1≤x≤1|21–n T

n (x)|that has the least value, the maximum being equal to21 –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– x2)y  xx–3xy 

x + n(n +2)y =0 and are defined by the formulas

U n (x) = sin[(n +1) arccos x]

√

1– x2 =

2n (n +1)!

(2n+1)!

1

√

1– x2

d n

dx n(1– x2)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

U0(x) =1, U1(x) =2x, U2(x) =4x2–1, U3(x) =8x3–4x, U4(x) =16x4–12x2+1 The recurrence formulas:

U n+1(x) =2xU n (x) – U n–1(x), n≥ 2

The generating function is

1

1–2sx + s2 =

∞



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).

18.17.2-3 Chebyshev functions of the second kind.

The Chebyshev functions of the second kind,

U0(x) = arcsin x,

Un (x) = sin(n arccos x) =

√

1– x2

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

U0(x) =0, U1(x) =

√

1– x2, U2(x) =2x √

1– x2,

U3(x) = (4 x2–1)√1– x2, U5(x) = (8 x3–4x)√

1– x2 The recurrence formulas:

Un+1(x) =2xUn (x) –Un–1(x), n≥ 2

The functionsUn (x) form an orthogonal system on the interval –1 < x <1, with

 1

– 1

Un √ (x)Um (x)

1– x2 dx=

0 if n≠m or n = m =0,

1

2π if n = m≠ 0

The generating function is

√

1– x2

1–2sx + s2 =

∞



n=0

Un+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) nexp x2 d n

dx nexp –x2

=

[n/2 ]

m=0

(–1)m n!

m ! (n –2m)!(2x)n–2m The first five polynomials are

H0(x) =1, H1(x) =2x, H2(x) =4x2–2, H3(x) =8x3–12x, H4(x) =16x4–48x2+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:

H2n (x) = (–1)n22n+1

√

π exp x2  ∞

0 exp –t

2

t2ncos(2xt ) dt,

H2n+1(x) = (–1)n22n+2

√

π exp x2  ∞

0 exp –t

2

t2n+1sin(2xt ) dt, where n =0, 1,2,

18.17.3-2 Orthogonality The generating function An asymptotic formula.

The functions H n (x) form an orthogonal system on the interval – ∞<x<∞ with weight e–x2

:

 ∞

–∞exp –x

2

H n (x)H m (x) dx =



0 if n≠m,

√

π2n n ! if n = m.

Generating function:

exp –s2+2sx

=

∞



n=0

H n (x) s

n

n!.

Asymptotic formula as n → ∞:

H n (x)≈ 2n+21n n2e–n2 exp x2

cos√

2n+1x– 12πn



18.17.3-3 Hermite functions

The Hermite functions h n (x) are introduced by the formula

h n (x) = exp



–1

2x2



H n (x) = (–1) nexp1

2x2

 d n

dx n exp –x2

, n=0,1,2, The Hermite functions satisfy the second-order linear ordinary differential equation

h 

xx+ (2n+1– x2)h =0

The functions h n (x) form an orthogonal system on the interval – ∞ < x < ∞ with

∞

–∞ h n (x)h m (x) dx =



0 if n≠m,

√

π2n n ! if n = m.

18.17.4 Jacobi Polynomials and Gegenbauer Polynomials

18.17.4-1 Jacobi polynomials

The Jacobi polynomials, P n α,β (x), are solutions of the second-order linear ordinary

differ-ential equation

(1– x2)y xx  +

β – α – (α + β +2)xy 

x + n(n + α + β +1)y =0 and are defined by the formulas

P α,β

n (x) = (–1)n

2n n!(1– x)–α(1+ x)–β d n

dx n

*

(1– x) α+n(1+ x) β+n+

=2–nn

m=0

C m n+α C n+β n–m (x –1)n–m (x +1)m, where the C b aare binomial coefficients

The generating function:

2α+β R– 1(1– s + R)–α(1+ s + R)–β =

∞



n=0

P α,β

n (x)s n, R=

√

1–2xs + s2, |s|<1 The Jacobi polynomials are orthogonal on the interval –1 ≤x≤ 1with weight (1–x)α(1+x)β:

 1

– 1 ( 1– x) α( 1+ x) β P α,β

n (x)P m α,β (x) dx =

⎧

⎨

⎩

2α+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– x2)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– x2)–λ+1 2 d n

dx n(1– x2)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 n(λ) (x) – n+2λ–1

n+1 C n–(λ)1(x);

C(λ)

n (–x) = (–1) n C(λ)

n (x), dx d C n(λ) (x) =2λC(λ+1 )

n–1 (x).

The generating function:

1 (1–2xs + s2)λ =

∞



n=0

C(λ)

n (x)s n.

The Gegenbauer polynomials are orthogonal on the interval –1 ≤ x ≤ 1 with weight (1– x2)λ–1 2:

 1

– 1(1– x2)λ–1 2C(λ)

n (x)C m(λ) (x) dx =

⎧

⎨

⎩

πΓ(2λ + n)

22λ–1(λ + n)n!Γ2(λ) if n = m.

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 n k 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

B0(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

B0(x) =1, B1(x) = x – 12, B2(x) = x2– x + 16, B3(x) = x3– 32x2+ 1

2x,

B4(x) = x4–2x3+ x2– 1

30, B5(x) = x5– 52x4+ 53x3– 16x. Basic properties:

B n (x +1) – Bn (x) = nx n–1, B 

n+1(x) = (n +1)Bn (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, and n =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

Fourier series expansions:

B n (x) = –2(2n!

π)n

∞



k=1

cos(2πkx– 12πn)

k n , (n =1, 0< x <1; n >1, 0 ≤x≤ 1);

B2n–1(x) =2(–1)n(2n–1)!

(2π)2n–1

∞



k=1

sin(2kπx)

k2n–1 (n =1, 0< x <1; n >1, 0 ≤x≤ 1);

B2n (x) =2(–1)n (2n)!

(2π)2n

∞



k=1

cos(2kπx)

k2n (n =1,2, , 0 ≤x≤ 1)

Integrals:  x

a B n (t) dt =

B n+1(x) – B n+1(a)

 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 nare Bernoulli numbers

18.18.2 Euler Polynomials

18.18.2-1 Definition Basic properties

Definition:

E n (x) =

n



k=0

C k

n E2n k



x– 1 2

n–k

(n =0,1, 2, ),

where C n k are the binomial coefficients and E nare Euler numbers

The first six Euler polynomials are given by

E0(x) =1, E1(x) = x – 12, E2(x) = x2– x, E3(x) = x3– 32x2+ 1

4,

E4(x) = x4–2x3+ x, E

5(x) = x5– 52x4+ 5

2x2– 12. Basic properties:

E n (x +1) + En (x) =2x n , E 

n+1= (n +1)En (x),

E n(1– x) = (–1) n E

n (x), (–1)n+1E

n (–x) = E n (x) –2x n, where the prime denotes a derivative with respect to x, and n =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+1m 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,