Szegő recurrence relations

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1: 16.25 Methods of Computation

… ►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. …There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). …In these cases integration, or recurrence, in either a forward or a backward direction is unstable. …

2: 24.14 Sums

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§24.14(i) Quadratic Recurrence Relations

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§24.14(ii) Higher-Order Recurrence Relations

… ►These identities can be regarded as higher-order recurrences. …

3: 33.17 Recurrence Relations and Derivatives

§33.17 Recurrence Relations and Derivatives

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4: 24.5 Recurrence Relations

§24.5 Recurrence Relations

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§24.5(i) Basic Relations

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5: 8.25 Methods of Computation

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§8.25(v) Recurrence Relations

… ►An efficient procedure, based partly on the recurrence relations (8.8.5) and (8.8.6), is described in Gautschi (1979b, 1999). …

6: 10.29 Recurrence Relations and Derivatives

§10.29 Recurrence Relations and Derivatives

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§10.29(i) Recurrence Relations

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7: 10.51 Recurrence Relations and Derivatives

§10.51 Recurrence Relations and Derivatives

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$$n{f}_{n-1}(z)-(n+1)f_{n+1}(z)=(2n+1)f_n^{\prime }(z),$$ $n=1,2,\mathrm{\dots }$,

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$$n{g}_{n-1}(z)+(n+1)g_{n+1}(z)=(2n+1)g_n^{\prime }(z),$$ $n=1,2,\mathrm{\dots }$,

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8: 18.9 Recurrence Relations and Derivatives

§18.9 Recurrence Relations and Derivatives

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§18.9(i) Recurrence Relations

… ►with initial values $p_0(x)=1$ and $p_1(x)=A_{0}x+B_0$. … ►with initial values $p_0(x)=1$ and $p_1(x)=a_0^{-1}(x-b_0)$. … ►and the structure relation …

9: 33.4 Recurrence Relations and Derivatives

§33.4 Recurrence Relations and Derivatives

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10: 33.23 Methods of Computation

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§33.23(iv) Recurrence Relations

►In a similar manner to §33.23(iii) the recurrence relations of §§33.4 or 33.17 can be used for a range of values of the integer $\mathrm{\ell }$, provided that the recurrence is carried out in a stable direction (§3.6). … ►Noble (2004) obtains double-precision accuracy for $W_{-\eta ,\mu }\left(2\rho \right)$ for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7). …