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1: 1.13 Differential Equations
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§1.13(vii) Closed-Form Solutions
… ►§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms
… ►This is the Sturm-Liouville form of a second order differential equation, where ′ denotes . … ►Transformation to Liouville normal Form
►Equation (1.13.26) with may be transformed to the Liouville normal form …2: 20 Theta Functions
Chapter 20 Theta Functions
…3: 26.3 Lattice Paths: Binomial Coefficients
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is the number of ways of choosing objects from a collection of distinct objects without regard to order.
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§26.3(ii) Generating Functions
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26.3.4
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§26.3(v) Limiting Form
…4: 26.5 Lattice Paths: Catalan Numbers
5: 26.10 Integer Partitions: Other Restrictions
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►The set is denoted by .
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§26.10(v) Limiting Form
… ►§26.10(vi) Bessel-Function Expansion
… ►where is the modified Bessel function (§10.25(ii)), and …The quantity is real-valued. …6: 10.1 Special Notation
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►The main functions treated in this chapter are the Bessel functions
, ; Hankel functions
, ; modified Bessel functions
, ; spherical Bessel functions
, , , ; modified spherical Bessel functions
, , ; Kelvin functions
, , , .
For the spherical Bessel functions and modified spherical Bessel functions the order is a nonnegative integer.
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►A common alternative notation for is .
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►Abramowitz and Stegun (1964): , , , , for , , , , respectively, when .
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►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
7: 8.17 Incomplete Beta Functions
§8.17 Incomplete Beta Functions
… ►where, as in §5.12, denotes the beta function: … ►Addendum: For a companion equation see (8.17.24). … ►For a historical profile of see Dutka (1981). … ►For the hypergeometric function see §15.2(i). …8: 26.9 Integer Partitions: Restricted Number and Part Size
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►A useful representation for a partition is the Ferrers graph in which the integers in the partition are each represented by a row of dots.
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►The conjugate partition is obtained by reflecting the Ferrers graph across the main diagonal or, equivalently, by representing each integer by a column of dots.
…Conjugation establishes a one-to-one correspondence between partitions of into at most parts and partitions of into parts with largest part less than or equal to .
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►Equations (26.9.2)–(26.9.3) are examples of closed forms that can be computed explicitly for any positive integer .
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§26.9(iv) Limiting Form
…9: Bibliography D
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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann.
Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).
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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire
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Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter.
SIAM J. Math. Anal. 21 (4), pp. 995–1018.
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10: Bibliography R
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Normal limit theorems for symmetric random matrices.
Probab. Theory Related Fields 112 (3), pp. 411–423.
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A code to calculate (high order) Bessel functions based on the continued fractions method.
Comput. Phys. Comm. 76 (3), pp. 381–388.
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On the definition and properties of generalized - symbols.
J. Math. Phys. 20 (12), pp. 2398–2415.
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Applied Bessel Functions.
Dover Publications Inc., New York.
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On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media.
In Differential Operators and Related Topics, Vol. I (Odessa,
1997),
Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.
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