infinite%20products
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1: 27.2 Functions
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►Functions in this section derive their properties from the fundamental
theorem of arithmetic, which states that every integer can be represented uniquely as a product of prime powers,
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27.2.1
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►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
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►It is the special case of the function that counts the number of ways of expressing as the product of factors, with the order of factors taken into account.
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2: Bibliography R
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Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions.
ACM Trans. Math. Softw. 40 (2), pp. 14:1–14:12.
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Integral representations for products of Airy functions. II. Cubic products.
Z. Angew. Math. Phys. 48 (4), pp. 646–655.
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Integral representations for products of Airy functions. III. Quartic products.
Z. Angew. Math. Phys. 48 (4), pp. 656–664.
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Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function.
Mathematics 9 (16).
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Sources in the development of mathematics.
Cambridge University Press, Cambridge.
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3: Bibliography D
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Sums of products of Bernoulli numbers.
J. Number Theory 60 (1), pp. 23–41.
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Infinite integrals in the theory of Bessel functions.
Quart. J. Math., Oxford Ser. 1 (1), pp. 122–145.
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Vector coupling coefficients as products of prime factors.
Comput. Phys. Comm. 4 (2), pp. 268–274.
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results.
SIAM J. Math. Anal. 9 (1), pp. 76–86.
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4: Bibliography S
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A property of the zeros of cross-product Bessel functions of different orders.
Z. Angew. Math. Mech. 56 (2), pp. 120–121.
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On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series.
J. Indian Math. Soc. (N. S.) 4, pp. 34–38.
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Uniform asymptotic forms of modified Mathieu functions.
Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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The Laplace transforms of products of Airy functions.
Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.
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Computation of infinite integrals involving Bessel functions of arbitrary order by the -transformation.
J. Comput. Appl. Math. 78 (1), pp. 125–130.
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5: Bibliography C
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On the representation of a large even integer as the sum of a prime and the product of at most two primes.
Kexue Tongbao (Foreign Lang. Ed.) 17, pp. 385–386.
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Remarks on the zeros of cross-product Bessel functions.
J. Soc. Indust. Appl. Math. 12 (3), pp. 580–587.
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The analyticity of cross-product Bessel function zeros.
Proc. Cambridge Philos. Soc. 62, pp. 215–226.
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The asymptotic nature of zeros of cross-product Bessel functions.
Quart. J. Mech. Appl. Math. 19 (4), pp. 511–522.
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Product formulas and convolutions for angular and radial spheroidal wave functions.
Trans. Amer. Math. Soc. 338 (2), pp. 695–710.
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6: Bibliography I
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The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of and of Bessel functions of any real order
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Linear Algebra Appl. 194, pp. 35–70.
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Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines.
Z. Angew. Math. Mech. 75 (12), pp. 917–926.
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The real roots of Bernoulli polynomials.
Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
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On polynomials orthogonal with respect to certain Sobolev inner products.
J. Approx. Theory 65 (2), pp. 151–175.
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7: Bibliography
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Integrals of products of Bernoulli polynomials.
J. Math. Anal. Appl. 381 (1), pp. 10–16.
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Integrals of products of Airy functions.
J. Phys. A 10 (4), pp. 485–490.
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There are infinitely many Carmichael numbers.
Ann. of Math. (2) 139 (3), pp. 703–722.
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Table of Definite and Infinite Integrals.
Physical Sciences Data, Vol. 13, Elsevier Scientific Publishing Co., Amsterdam.
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An integral of products of ultraspherical functions and a -extension.
J. London Math. Soc. (2) 33 (1), pp. 133–148.
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8: Bibliography W
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Infinitely differentiable generalized logarithmic and exponential functions.
Math. Comp. 57 (196), pp. 723–733.
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Reduction formulae for products of theta functions.
J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.
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The Nahm equations, finite-gap potentials and Lamé functions.
J. Phys. A 20 (10), pp. 2679–2683.
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The generalised product moment distribution in samples from a normal multivariate population.
Biometrika 20A, pp. 32–52.
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9: Bibliography V
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Integrating products of Bessel functions with an additional exponential or rational factor.
Comput. Phys. Comm. 178 (8), pp. 578–590.
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An infinite series of Weber’s parabolic cylinder functions.
Proc. Benares Math. Soc. (N.S.) 3, pp. 37.
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Some novel infinite series of spherical Bessel functions.
Quart. Appl. Math. 42 (3), pp. 321–324.
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Expansions in products of Heine-Stieltjes polynomials.
Constr. Approx. 15 (4), pp. 467–480.
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Integral representations for products of Lamé functions by use of fundamental solutions.
SIAM J. Math. Anal. 15 (3), pp. 559–569.
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