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21: 10.63 Recurrence Relations and Derivatives

§10.63 Recurrence Relations and Derivatives

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§10.63(i) ${\mathrm{ber}}_{\nu }x$, ${\mathrm{bei}}_{\nu }x$, ${\ker }_{\nu }x$, ${\mathrm{kei}}_{\nu }x$

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22: 10.74 Methods of Computation

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§10.74(iv) Recurrence Relations

►If values of the Bessel functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, or the other functions treated in this chapter, are needed for integer-spaced ranges of values of the order $\nu$, then a simple and powerful procedure is provided by recurrence relations typified by the first of (10.6.1). … ►Then $J_n\left(x\right)$ and $Y_n\left(x\right)$ can be generated by either forward or backward recurrence on $n$ when , but if $n>x$ then to maintain stability $J_n\left(x\right)$ has to be generated by backward recurrence on $n$, and $Y_n\left(x\right)$ has to be generated by forward recurrence on $n$. …

23: 26.3 Lattice Paths: Binomial Coefficients

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§26.3(iii) Recurrence Relations

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24: 26.7 Set Partitions: Bell Numbers

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§26.7(iii) Recurrence Relation

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25: 18.33 Polynomials Orthogonal on the Unit Circle

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§18.33(ii) Recurrence Relations

… ►For an alternative and more detailed approach to the recurrence relations, see §18.33(vi). … ►

Recurrence Relations

… ►Equivalent to the recurrence relations (18.33.23), (18.33.24) are the inverse Szegő recurrence relations … ►while combination of (18.33.27) and (18.33.23) gives the three-term recurrence relation …

26: 11.13 Methods of Computation

… ►In consequence forward recurrence, backward recurrence, or boundary-value methods may be necessary. …

27: 24.19 Methods of Computation

… ►Equations (24.5.3) and (24.5.4) enable $B_n$ and $E_n$ to be computed by recurrence. …For example, the tangent numbers $T_n$ can be generated by simple recurrence relations obtained from (24.15.3), then (24.15.4) is applied. …

28: 14.21 Definitions and Basic Properties

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§14.21(iii) Properties

… ►This includes, for example, the Wronskian relations (14.2.7)–(14.2.11); hypergeometric representations (14.3.6)–(14.3.10) and (14.3.15)–(14.3.20); results for integer orders (14.6.3)–(14.6.5), (14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11)–(14.7.16); behavior at singularities (14.8.7)–(14.8.16); connection formulas (14.9.11)–(14.9.16); recurrence relations (14.10.3)–(14.10.7). …

29: 26.9 Integer Partitions: Restricted Number and Part Size

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§26.9(iii) Recurrence Relations

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30: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

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§26.4(iii) Recurrence Relation

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