relation to power series
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21: 24.19 Methods of Computation
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►Equations (24.5.3) and (24.5.4) enable and
to be computed by recurrence.
…For example, the tangent numbers can be generated by simple recurrence relations obtained from (24.15.3), then (24.15.4) is applied.
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►For number-theoretic applications it is important to compute for ; in particular to find the irregular pairs
for which .
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Tanner and Wagstaff (1987) derives a congruence for Bernoulli numbers in terms of sums of powers. See also §24.10(iii).
22: 10.74 Methods of Computation
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►The power-series expansions given in §§10.2 and 10.8, together with the connection formulas of §10.4, can be used to compute the Bessel and Hankel functions when the argument or is sufficiently small in absolute value.
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►In other circumstances the power series are prone to slow convergence and heavy numerical cancellation.
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►A comprehensive and powerful approach is to integrate the differential equations (10.2.1) and (10.25.1) by direct numerical methods.
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►In the interval , needs to be integrated in the forward direction and in the backward direction, with initial values for the former obtained from the power-series expansion (10.2.2) and for the latter from asymptotic expansions (§§10.17(i) and 10.20(i)).
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►If values of the Bessel functions , , or the other functions treated in this chapter, are needed for integer-spaced ranges of values of the order , then a simple and powerful procedure is provided by recurrence relations typified by the first of (10.6.1).
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23: Bibliography B
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A Brief Introduction to Theta Functions.
Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York.
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Formal Power Series and Algebraic Combinatorics.
DIMACS Series in Discrete Mathematics and Theoretical Computer
Science, Vol. 24, American Mathematical Society, Providence, RI.
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Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind.
U.S. Government Printing Office, Washington, D.C..
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Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind.
U.S. Government Printing Office, Washington, D.C..
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Concerning the zeros of some functions related to Bessel functions.
J. Mathematical Phys. 10 (9), pp. 1729–1744.
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24: 27.4 Euler Products and Dirichlet Series
§27.4 Euler Products and Dirichlet Series
… ►if the series on the left is absolutely convergent. … ►Euler products are used to find series that generate many functions of multiplicative number theory. … ►called Dirichlet series with coefficients . …The following examples have generating functions related to the zeta function: …25: 27.14 Unrestricted Partitions
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►Multiplying the power series for with that for and equating coefficients, we obtain the recursion formula
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§27.14(iv) Relation to Modular Functions
… ►This is related to the function in (27.14.2) by … ►The 24th power of in (27.14.12) with is an infinite product that generates a power series in with integer coefficients called Ramanujan’s tau function : …The tau function is multiplicative and satisfies the more general relation: …26: 18.2 General Orthogonal Polynomials
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►The orthogonality relations (18.2.1)–(18.2.3) each determine the polynomials uniquely up to constant factors, which may be fixed by suitable standardizations.
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►However, if OP’s have an orthogonality relation on a bounded interval, then their orthogonality measure is unique, up to a positive constant factor.
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►For such a system, functions and sequences () satisfying can be related to each other in a similar way as was done for Fourier series in (1.8.1) and (1.8.2):
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►where and are formal power series in , with , and .
…If is the formal power series such that then a property equivalent to (18.2.45) with is that
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27: 8.21 Generalized Sine and Cosine Integrals
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§8.21(v) Special Values
… ►§8.21(vi) Series Expansions
►Power-Series Expansions
… ►Spherical-Bessel-Function Expansions
… ►For (8.21.16), (8.21.17), and further expansions in series of Bessel functions see Luke (1969b, pp. 56–57). …28: 27.10 Periodic Number-Theoretic Functions
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►Every function periodic (mod ) can be expressed as a finite Fourier
series of the form
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27.10.2
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►This is the sum of the th powers of the primitive th roots of unity.
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►is a periodic function of and has the finite Fourier-series expansion
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►It is defined by the relation
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29: 13.29 Methods of Computation
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►A comprehensive and powerful approach is to integrate the differential equations (13.2.1) and (13.14.1) by direct numerical methods.
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§13.29(iv) Recurrence Relations
►The recurrence relations in §§13.3(i) and 13.15(i) can be used to compute the confluent hypergeometric functions in an efficient way. … ►normalizing relation … ►normalizing relation …30: Bibliography C
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Power series for inverse Jacobian elliptic functions.
Math. Comp. 77 (263), pp. 1615–1621.
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Introduction to the Theory of Fourier’s Series and Integrals.
3rd edition, Macmillan, London.
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A Unified Approach to Recurrence Algorithms.
In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.),
International Series of Computational Mathematics, Vol. 119, pp. 97–120.
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Numerical Grid Methods and Their Application to Schrödinger’s Equation.
NATO Advanced Science Institutes Series C: Mathematical and
Physical Sciences, Vol. 412, Kluwer Academic Publishers, Dordrecht.
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Performance evaluation of programs related to the real gamma function.
ACM Trans. Math. Software 17 (1), pp. 46–54.
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