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1: 23.2 Definitions and Periodic Properties

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§23.2(i) Lattices

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§23.2(ii) Weierstrass Elliptic Functions

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§23.2(iii) Periodicity

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2: 22.15 Inverse Functions

§22.15 Inverse Functions

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§22.15(i) Definitions

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§22.15(ii) Representations as Elliptic Integrals

… ►For representations of the inverse functions as symmetric elliptic integrals see §19.25(v). …

3: 22.2 Definitions

§22.2 Definitions

… ►where $K\left(k\right)$, $K^{\prime }\left(k\right)$ are defined in §19.2(ii). … ►As a function of $z$, with fixed $k$, each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. … … ► $\mathrm{s}\mathrm{s}(z,k)=1$. …

4: 19.16 Definitions

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§19.16(i) Symmetric Integrals

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§19.16(ii) $R_{-a}(𝐛;𝐳)$

… ►The $R$-function is often used to make a unified statement of a property of several elliptic integrals. … ►For generalizations and further information, especially representation of the $R$-function as a Dirichlet average, see Carlson (1977b). … ►

§19.16(iii) Various Cases of $R_{-a}(𝐛;𝐳)$

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5: Wolter Groenevelt

… ►Groenevelt’s research interests is in special functions and orthogonal polynomials and their relations with representation theory and interacting particle systems. ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. ►In July 2023, Groenevelt was named Contributing Developer of the NIST Digital Library of Mathematical Functions.

6: 23 Weierstrass Elliptic and Modular
Functions

Chapter 23 Weierstrass Elliptic and Modular Functions

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7: 8.17 Incomplete Beta Functions

§8.17 Incomplete Beta Functions

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§8.17(ii) Hypergeometric Representations

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§8.17(iii) Integral Representation

… ►Further integral representations can be obtained by combining the results given in §8.17(ii) with §15.6. … ►

§8.17(vi) Sums

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8: 20 Theta Functions

Chapter 20 Theta Functions

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9: 25.11 Hurwitz Zeta Function

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§25.11(iii) Representations by the Euler–Maclaurin Formula

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§25.11(iv) Series Representations

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§25.11(vii) Integral Representations

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§25.11(viii) Further Integral Representations

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§25.11(x) Further Series Representations

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10: 14.19 Toroidal (or Ring) Functions

§14.19 Toroidal (or Ring) Functions

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§14.19(i) Introduction

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§14.19(ii) Hypergeometric Representations

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§14.19(iii) Integral Representations

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§14.19(v) Whipple’s Formula for Toroidal Functions

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