recursion formulas

(0.002 seconds)

11—14 of 14 matching pages

11: 12.14 The Function $W\left(a,x\right)$

… ►

§12.14(iv) Connection Formula

… ►

12.14.9

w_{1}(a,x)=\sum_{n=0}^{\infty}\alpha_{n}(a)\frac{x^{2n}}{(2n)!},

ⓘ

Defines:: $w_{1}(a,x)$ : even solution (locally)
Symbols:: $!$ : factorial (as in $n!$ ), $x$ : real variable, $n$ : nonnegative integer, $a$ : real or complex parameter and $\alpha_{n}(a)$ : recursion
A&S Ref:: 19.16.1 (modification of)
Permalink:: http://dlmf.nist.gov/12.14.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §12.14(v), §12.14 and Ch.12

►

12.14.10

w_{2}(a,x)=\sum_{n=0}^{\infty}\beta_{n}(a)\frac{x^{2n+1}}{(2n+1)!},

ⓘ

Defines:: $w_{2}(a,x)$ : odd solution (locally)
Symbols:: $!$ : factorial (as in $n!$ ), $x$ : real variable, $n$ : nonnegative integer, $a$ : real or complex parameter and $\beta_{n}(a)$ : recursion
A&S Ref:: 19.16.2 (modification of)
Permalink:: http://dlmf.nist.gov/12.14.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §12.14(v), §12.14 and Ch.12

►where

\alpha_{n}(a)

and

\beta_{n}(a)

satisfy the recursion relations …

12: 18.30 Associated OP’s

… ►The recursion relation for the associated Laguerre polynomials, see (18.30.2), (18.30.3) is … ►The recursion relation for the associated Hermite polynomials, see (18.30.2), and (18.30.3), is … ►Defining associated orthogonal polynomials and their relationship to their corecursive counterparts is particularly simple via use of the recursion relations for the monic, rather than via those for the traditional polynomials. … ►In the monic case, the monic associated polynomials

\hat{p}_{n}(x;c)

of order

c

with respect to the

\hat{p}_{n}(x)

are obtained by simply changing the initialization and recursions, respectively, of (18.30.2) and (18.30.3) to … ►followed by use of the

c=0

recursion of (18.30.27). …

13: 29.6 Fourier Series

… ►This solution can be constructed from (29.6.4) by backward recursion, starting with

A_{2n+2}=0

and an arbitrary nonzero value of

A_{2n}

, followed by normalization via (29.6.5) and (29.6.6). …

14: 18.39 Applications in the Physical Sciences

… ►The discrete variable representations (DVR) analysis is simplest when based on the classical OP’s with their analytically known recursion coefficients (Table 3.5.17_5), or those non-classical OP’s which have analytically known recursion coefficients, making stable computation of the

x_{i}

and

w_{i}

, from the J-matrix as in §3.5(vi), straightforward. …Table 18.39.1 lists typical non-classical weight functions, many related to the non-classical Freud weights of §18.32, and §32.15, all of which require numerical computation of the recursion coefficients (i. … ►Following the method of Schwartz (1961), Yamani and Reinhardt (1975), Bank and Ismail (1985), and Ismail (2009, §5.8) have shown this is equivalent to determination of

x

such that

c_{N}(x)=0

in the recursion scheme …The recursion of (18.39.46) is that for the type 2 Pollaczek polynomials of (18.35.2), with

\lambda=l+1

a=-b=2Z/s

, and

c=0

, and terminates for

x=x^{N}_{i}

being a zero of the polynomial of order

N

. … ►As this follows from the three term recursion of (18.39.46) it is referred to as the J-Matrix approach, see (3.5.31), to single and multi-channel scattering numerics. …