# product representation

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## 11—20 of 53 matching pages

##### 11: 28.28 Integrals, Integral Representations, and Integral Equations
28.28.23 $\dfrac{2}{\pi}\int_{0}^{\pi}\mathcal{C}^{(j)}_{2\ell+2}(2hR)\sin\left((2\ell+2% )\phi\right)\mathrm{se}_{2m+2}\left(t,h^{2}\right)\mathrm{d}t=(-1)^{\ell+m}B^{% 2m+2}_{2\ell+2}(h^{2}){\mathrm{Ms}^{(j)}_{2m+2}}\left(z,h\right).$
###### §28.28(iv) Integrals of Products of Mathieu Functions of Integer Order
28.28.49 $\widehat{\alpha}_{n,m}^{(c)}=\frac{1}{2\pi}\int_{0}^{2\pi}\cos t\mathrm{ce}_{n% }\left(t,h^{2}\right)\mathrm{ce}_{m}\left(t,h^{2}\right)\mathrm{d}t=(-1)^{p+1}% \dfrac{2}{\mathrm{i}\pi}\dfrac{\mathrm{ce}_{n}\left(0,h^{2}\right)\mathrm{ce}_% {m}\left(0,h^{2}\right)}{h\mathrm{Dc}_{0}\left(n,m,0\right)}.$
##### 12: 13.25 Products
For integral representations, integrals, and series containing products of $M_{\kappa,\mu}\left(z\right)$ and $W_{\kappa,\mu}\left(z\right)$ see Erdélyi et al. (1953a, §6.15.3).
##### 13: 27.14 Unrestricted Partitions
27.14.2 $\mathit{f}\left(x\right)=\prod_{m=1}^{\infty}(1-x^{m})=\left(x;x\right)_{% \infty},$ $|x|<1$,
##### 14: 5.17 Barnes’ $G$-Function (Double Gamma Function)
5.17.3 $G\left(z+1\right)=(2\pi)^{z/2}\exp\left(-\tfrac{1}{2}z(z+1)-\tfrac{1}{2}\gamma z% ^{2}\right)\*\prod_{k=1}^{\infty}\left(\left(1+\frac{z}{k}\right)^{k}\exp\left% (-z+\frac{z^{2}}{2k}\right)\right).$
##### 15: 14.30 Spherical and Spheroidal Harmonics
For a series representation of the product of two Dirac deltas in terms of products of spherical harmonics see §1.17(iii). …
##### 16: 10.59 Integrals
For an integral representation of the Dirac delta in terms of a product of spherical Bessel functions of the first kind see §1.17(ii), and for a generalization see Maximon (1991). …
##### 17: 26.15 Permutations: Matrix Notation
The inversion number of $\sigma$ is a sum of products of pairs of entries in the matrix representation of $\sigma$: …
##### 18: 14.18 Sums
For a series representation of the Dirac delta in terms of products of Legendre polynomials see (1.17.22). …
##### 19: Bibliography C
• J. Chen (1966) 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.
• ##### 20: 10.22 Integrals
See also §1.17(ii) for an integral representation of the Dirac delta in terms of a product of Bessel functions. …