equivalent equation for contiguous functions

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21: 4.2 Definitions

… ► … ►Equivalently, … ►

§4.2(iii) The Exponential Function

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§4.2(iv) Powers

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22: 10.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►For the spherical Bessel functions and modified spherical Bessel functions the order $n$ is a nonnegative integer. For the other functions when the order $\nu$ is replaced by $n$, it can be any integer. For the Kelvin functions the order $\nu$ is always assumed to be real. … ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

23: 4.23 Inverse Trigonometric Functions

§4.23 Inverse Trigonometric Functions

… ►An equivalent definition is … ►With $k\in \mathbb{Z}$, the general solutions of the equations … ►Equivalently, … ►Equivalently, and again when , …

24: 4.37 Inverse Hyperbolic Functions

§4.37 Inverse Hyperbolic Functions

… ►An equivalent definition is … ►

Other Inverse Functions

… ►With $k\in \mathbb{Z}$, the general solutions of the equations … ►

§4.37(vi) Interrelations

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25: 16.2 Definition and Analytic Properties

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§16.2(i) Generalized Hypergeometric Series

… ►Equivalently, the function is denoted by ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$ or ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$, and sometimes, for brevity, by ${}_p{}^{}F_q^{}\left(z\right)$. … ► … ►

Polynomials

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§16.2(v) Behavior with Respect to Parameters

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26: 12.14 The Function $W(a,x)$

§12.14 The Function $W(a,x)$

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§12.14(i) Introduction

… ►Equivalently, … ►The differential equation … ►For properties of the modulus and phase functions, including differential equations and asymptotic expansions for large $x$, see Miller (1955, pp. 87–88). …

27: 23.2 Definitions and Periodic Properties

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§23.2(i) Lattices

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§23.2(ii) Weierstrass Elliptic Functions

… ► … ►Hence $\mathrm{\wp }\left(z\right)$ is an elliptic function, that is, $\mathrm{\wp }\left(z\right)$ is meromorphic and periodic on a lattice; equivalently, $\mathrm{\wp }\left(z\right)$ is meromorphic and has two periods whose ratio is not real. …

28: 8.17 Incomplete Beta Functions

§8.17 Incomplete Beta Functions

… ►Addendum: For a companion equation see (8.17.24). … ►

§8.17(ii) Hypergeometric Representations

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§8.17(iii) Integral Representation

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§8.17(vi) Sums

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29: 11.10 Anger–Weber Functions

§11.10 Anger–Weber Functions

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§11.10(ii) Differential Equations

►The Anger and Weber functions satisfy the inhomogeneous Bessel differential equation … ►

§11.10(vi) Relations to Other Functions

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30: 30.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►Meixner and Schäfke (1954) use $\mathrm{ps}$, $\mathrm{qs}$, $\mathrm{Ps}$, $\mathrm{Qs}$ for $\mathrm{𝖯𝗌}$, $\mathrm{𝖰𝗌}$, $\mathrm{𝑃𝑠}$, $\mathrm{𝑄𝑠}$, respectively. ►

Other Notations

►Flammer (1957) and Abramowitz and Stegun (1964) use ${\lambda }_{mn}(\gamma )$ for ${\lambda }_n^m\left({\gamma }^2\right)+{\gamma }^2$, $R_{mn}^{(j)}(\gamma ,z)$ for $S_n^{m(j)}(z,\gamma )$, and …