# recurrence

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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: 5.21 Methods of Computation
An effective way of computing $\Gamma\left(z\right)$ in the right half-plane is backward recurrence, beginning with a value generated from the asymptotic expansion (5.11.3). Or we can use forward recurrence, with an initial value obtained e. …For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). …
##### 3: 24.14 Sums
###### §24.14(ii) Higher-Order Recurrence Relations
These identities can be regarded as higher-order recurrences. …
##### 6: 8.25 Methods of Computation
###### §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). …
##### 7: 33.23 Methods of Computation
###### §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 $\ell$, 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). …
##### 9: 10.51 Recurrence Relations and Derivatives
###### §10.51 Recurrence Relations and Derivatives
$nf_{n-1}(z)-(n+1)f_{n+1}(z)=(2n+1)f_{n}^{\prime}(z),$ $n=1,2,\dots$,
$ng_{n-1}(z)+(n+1)g_{n+1}(z)=(2n+1)g_{n}^{\prime}(z),$ $n=1,2,\dotsc$,