general properties

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21: 10.21 Zeros

… ►For properties, computation, and generalizations see Kapitsa (1951b), Kerimov (1999, 2008), and Gupta and Muldoon (2000). …

22: 10.49 Explicit Formulas

… ►For a survey of properties of these polynomials and their generalizations see Grosswald (1978). …

23: 16.4 Argument Unity

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Watson’s Sum

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24: 23.15 Definitions

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§23.15(i) General Modular Functions

… ►A modular function $f(\tau )$ is a function of $\tau$ that is meromorphic in the half-plane $\mathrm{\Im }\tau >0$, and has the property that for all $𝒜\in \text{SL}(2,\mathbb{Z})$, or for all $𝒜$ belonging to a subgroup of SL$(2,\mathbb{Z})$, … …

25: 18.35 Pollaczek Polynomials

… ►For type 3 orthogonality (18.35.5) generalizes to …

26: 24.16 Generalizations

§24.16 Generalizations

… ►For this and other properties see Milne-Thomson (1933, pp. 126–153) or Nörlund (1924, pp. 144–162). … ►

§24.16(ii) Character Analogs

… ►For further properties see Berndt (1975a). ►

§24.16(iii) Other Generalizations

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27: 10.43 Integrals

… ►For further properties of the Bickley function, including asymptotic expansions and generalizations, see Amos (1983c, 1989) and Luke (1962, Chapter 8). …

28: 10.41 Asymptotic Expansions for Large Order

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§10.41(iv) Double Asymptotic Properties

►The series (10.41.3)–(10.41.6) can also be regarded as generalized asymptotic expansions for large $|z|$. … ►Moreover, because of the uniqueness property of asymptotic expansions (§2.1(iii)) this expansion must agree with (10.40.2), with $z$ replaced by $\nu z$, up to and including the term in $z^{-(\mathrm{\ell }-1)}$. … ►

§10.41(v) Double Asymptotic Properties (Continued)

… ►We first prove that for the expansions (10.20.6) for the Hankel functions $H_{\nu }^{(1)}\left(\nu z\right)$ and $H_{\nu }^{(2)}\left(\nu z\right)$ the $z$-asymptotic property applies when $z\to \pm \mathrm{i}\mathrm{\infty }$, respectively. …

29: 18.34 Bessel Polynomials

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§18.34(i) Definitions and Recurrence Relation

… ►Often only the polynomials (18.34.2) are called Bessel polynomials, while the polynomials (18.34.1) and (18.34.3) are called generalized Bessel polynomials. … ►

§18.34(ii) Orthogonality

… ►See Ismail (2009, (4.10.9)) for orthogonality on the unit circle for general values of $a$. ►

§18.34(iii) Other Properties

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30: 3.11 Approximation Techniques

… ►The $c_n$ in (3.11.11) can be calculated from (3.11.10), but in general it is more efficient to make use of the orthogonal property (3.11.9). …