# double products

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## 1—10 of 27 matching pages

##### 1: 20.5 Infinite Products and Related Results
###### §20.5(iii) DoubleProducts
These double products are not absolutely convergent; hence the order of the limits is important. …
##### 2: 23.2 Definitions and Periodic Properties
The double series and double product are absolutely and uniformly convergent in compact sets in $\mathbb{C}$ that do not include lattice points. …
##### 3: 27.5 Inversion Formulas
27.5.8 $g(n)=\prod_{d\mathbin{|}n}f(d)\Longleftrightarrow f(n)=\prod_{d\mathbin{|}n}% \left(g\left(\frac{n}{d}\right)\right)^{\mu\left(d\right)}.$
##### 4: 34.4 Definition: $\mathit{6j}$ Symbol
The $\mathit{6j}$ symbol is defined by the following double sum of products of $\mathit{3j}$ symbols: …
##### 6: 5.17 Barnes’ $G$-Function (Double Gamma Function)
5.17.3 $G\left(z+1\right)=(2\pi)^{z/2}\exp\left(-\tfrac{1}{2}z(z+1)-\tfrac{1}{2}\gamma z% ^{2}\right)\*\prod_{k=1}^{\infty}\left(\left(1+\frac{z}{k}\right)^{k}\exp\left% (-z+\frac{z^{2}}{2k}\right)\right).$
##### 7: 16.14 Partial Differential Equations
In addition to the four Appell functions there are $24$ other sums of double series that cannot be expressed as a product of two ${{}_{2}F_{1}}$ functions, and which satisfy pairs of linear partial differential equations of the second order. …
##### 8: Mathematical Introduction
In addition, there is a comprehensive account of the great variety of analytical methods that are used for deriving and applying the extremely important asymptotic properties of the special functions, including double asymptotic properties (Chapter 2 and §§10.41(iv), 10.41(v)). …
 $\mathbb{C}$ complex plane (excluding infinity). … unity. … implies. is equivalent to. … double factorial: $2\cdot 4\cdot 6\cdots n$ if $n=2,4,6,\dotsc$; $1\cdot 3\cdot 5\cdots n$ if $n=1,3,5,\dotsc$; 1 if $n=0,-1$. …
##### 9: Bibliography R
• Yu. L. Ratis and P. Fernández de Córdoba (1993) A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.
• W. H. Reid (1995) Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.
• W. H. Reid (1997a) Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.
• W. H. Reid (1997b) Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.
• M. D. Rogers (2005) Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.
• ##### 10: Bibliography I
• M. Ikonomou, P. Köhler, and A. F. Jacob (1995) 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.
• IMSL (commercial C, Fortran, and Java libraries)
• A. Iserles, P. E. Koch, S. P. Nørsett, and J. M. Sanz-Serna (1991) On polynomials orthogonal with respect to certain Sobolev inner products. J. Approx. Theory 65 (2), pp. 151–175.