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1: 30.10 Series and Integrals
For product formulas and convolutions see Connett et al. (1993). …For expansions in products of spherical Bessel functions, see Flammer (1957, Chapter 6).
2: 27.5 Inversion Formulas
If a Dirichlet series F ( s ) generates f ( n ) , and G ( s ) generates g ( n ) , then the product F ( s ) G ( s ) generates
27.5.1 h ( n ) = d | n f ( d ) g ( n d ) ,
called the Dirichlet product (or convolution) of f and g . …
27.5.8 g ( n ) = d | n f ( d ) f ( n ) = d | n ( g ( n d ) ) μ ( d ) .
3: 2.6 Distributional Methods
We now derive an asymptotic expansion of 𝐼 μ f ( x ) for large positive values of x . In terms of the convolution product …The replacement of f ( t ) by its asymptotic expansion (2.6.9), followed by term-by-term integration leads to convolution integrals of the form …However, the left-hand side can be considered as the convolution of the two distributions associated with the functions t μ 1 and t s α , given by (2.6.12) and (2.6.13). … The method of distributions can be further extended to derive asymptotic expansions for convolution integrals: …
4: Bibliography G
  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
  • K. Girstmair (1990b) Dirichlet convolution of cotangent numbers and relative class number formulas. Monatsh. Math. 110 (3-4), pp. 231–256.
  • E. T. Goodwin (1949a) Recurrence relations for cross-products of Bessel functions. Quart. J. Mech. Appl. Math. 2 (1), pp. 72–74.
  • H. P. W. Gottlieb (1985) On the exceptional zeros of cross-products of derivatives of spherical Bessel functions. Z. Angew. Math. Phys. 36 (3), pp. 491–494.
  • D. P. Gupta and M. E. Muldoon (2000) Riccati equations and convolution formulae for functions of Rayleigh type. J. Phys. A 33 (7), pp. 1363–1368.
  • 5: Bibliography
  • M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.
  • T. Agoh and K. Dilcher (2011) Integrals of products of Bernoulli polynomials. J. Math. Anal. Appl. 381 (1), pp. 10–16.
  • J. R. Albright (1977) Integrals of products of Airy functions. J. Phys. A 10 (4), pp. 485–490.
  • W. L. Anderson (1982) Algorithm 588. Fast Hankel transforms using related and lagged convolutions. ACM Trans. Math. Software 8 (4), pp. 369–370.
  • R. Askey, T. H. Koornwinder, and M. Rahman (1986) An integral of products of ultraspherical functions and a q -extension. J. London Math. Soc. (2) 33 (1), pp. 133–148.
  • 6: 27.15 Chinese Remainder Theorem
    The Chinese remainder theorem states that a system of congruences x a 1 ( mod m 1 ) , , x a k ( mod m k ) , always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod m ), where m is the product of the moduli. … Their product m has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result ( mod m ) , which is correct to 20 digits. …
    7: Bibliography W
  • P. L. Walker (2012) Reduction formulae for products of theta functions. J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.
  • R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.
  • J. Wishart (1928) The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A, pp. 32–52.
  • R. Wong (1979) Explicit error terms for asymptotic expansions of Mellin convolutions. J. Math. Anal. Appl. 72 (2), pp. 740–756.
  • 8: 35.2 Laplace Transform
    Convolution Theorem
    If g j is the Laplace transform of f j , j = 1 , 2 , then g 1 g 2 is the Laplace transform of the convolution f 1 f 2 , where
    35.2.3 f 1 f 2 ( 𝐓 ) = 𝟎 < 𝐗 < 𝐓 f 1 ( 𝐓 𝐗 ) f 2 ( 𝐗 ) d 𝐗 .
    9: 2.5 Mellin Transform Methods
    with a < c < b . One of the two convolution integrals associated with the Mellin transform is of the form … In the half-plane z > max ( 0 , 2 ν ) , the product f ( 1 z ) h ( z ) has a pole of order two at each positive integer, and …
    2.5.29 I ( x ) = j , k = 1 2 I j k ( x ) ,
    2.5.31 I 21 ( x ) = 0 , for x 1 .
    10: Bibliography L
  • D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.
  • D. R. Lehman, W. C. Parke, and L. C. Maximon (1981) Numerical evaluation of integrals containing a spherical Bessel function by product integration. J. Math. Phys. 22 (7), pp. 1399–1413.
  • X. Li and R. Wong (1994) Error bounds for asymptotic expansions of Laplace convolutions. SIAM J. Math. Anal. 25 (6), pp. 1537–1553.
  • P. Linz and T. E. Kropp (1973) A note on the computation of integrals involving products of trigonometric and Bessel functions. Math. Comp. 27 (124), pp. 871–872.
  • S. K. Lucas (1995) Evaluating infinite integrals involving products of Bessel functions of arbitrary order. J. Comput. Appl. Math. 64 (3), pp. 269–282.