# integer parameters

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## 1—10 of 276 matching pages

##### 1: 14.34 Software
• Adams and Swarztrauber (1997). Integer parameters. Fortran.

• Braithwaite (1973). Integer parameters. Fortran.

• Delic (1979a). Integer parameters. Fortran.

• Gil and Segura (1998). Integer parameters. Fortran.

• Lozier and Smith (1981). Integer parameters. Fortran.

• ##### 2: 28.11 Expansions in Series of Mathieu Functions
28.11.3 $1=2\sum_{n=0}^{\infty}A_{0}^{2n}(q)\operatorname{ce}_{2n}\left(z,q\right),$
28.11.4 $\cos 2mz=\sum_{n=0}^{\infty}A_{2m}^{2n}(q)\operatorname{ce}_{2n}\left(z,q% \right),$ $m\neq 0$,
28.11.5 $\cos(2m+1)z=\sum_{n=0}^{\infty}A_{2m+1}^{2n+1}(q)\operatorname{ce}_{2n+1}\left% (z,q\right),$
28.11.6 $\sin(2m+1)z=\sum_{n=0}^{\infty}B_{2m+1}^{2n+1}(q)\operatorname{se}_{2n+1}\left% (z,q\right),$
28.11.7 $\sin(2m+2)z=\sum_{n=0}^{\infty}B_{2m+2}^{2n+2}(q)\operatorname{se}_{2n+2}\left% (z,q\right).$
##### 3: 28.23 Expansions in Series of Bessel Functions
28.23.6 ${\operatorname{Mc}^{(j)}_{2m}}\left(z,h\right)=(-1)^{m}\left(\operatorname{ce}% _{2m}\left(0,h^{2}\right)\right)^{-1}\sum_{\ell=0}^{\infty}(-1)^{\ell}A_{2\ell% }^{2m}(h^{2}){\cal C}_{2\ell}^{(j)}(2h\cosh z),$
28.23.7 ${\operatorname{Mc}^{(j)}_{2m}}\left(z,h\right)=(-1)^{m}\left(\operatorname{ce}% _{2m}\left(\tfrac{1}{2}\pi,h^{2}\right)\right)^{-1}\sum_{\ell=0}^{\infty}A_{2% \ell}^{2m}(h^{2}){\cal C}_{2\ell}^{(j)}(2h\sinh z),$
28.23.8 ${\operatorname{Mc}^{(j)}_{2m+1}}\left(z,h\right)=(-1)^{m}\left(\operatorname{% ce}_{2m+1}\left(0,h^{2}\right)\right)^{-1}\sum_{\ell=0}^{\infty}(-1)^{\ell}A_{% 2\ell+1}^{2m+1}(h^{2}){\cal C}_{2\ell+1}^{(j)}(2h\cosh z),$
28.23.9 ${\operatorname{Mc}^{(j)}_{2m+1}}\left(z,h\right)=(-1)^{m+1}\left(\operatorname% {ce}_{2m+1}'\left(\tfrac{1}{2}\pi,h^{2}\right)\right)^{-1}\coth z\sum_{\ell=0}% ^{\infty}(2\ell+1)A_{2\ell+1}^{2m+1}(h^{2}){\cal C}_{2\ell+1}^{(j)}(2h\sinh z),$
28.23.10 ${\operatorname{Ms}^{(j)}_{2m+1}}\left(z,h\right)=(-1)^{m}\left(\operatorname{% se}_{2m+1}'\left(0,h^{2}\right)\right)^{-1}\tanh z\sum_{\ell=0}^{\infty}(-1)^{% \ell}(2\ell+1)B_{2\ell+1}^{2m+1}(h^{2}){\cal C}_{2\ell+1}^{(j)}(2h\cosh z),$
##### 4: 28.19 Expansions in Series of $\operatorname{me}_{\nu+2n}$ Functions
28.19.4 $e^{\mathrm{i}\nu z}=\sum_{n=-\infty}^{\infty}c^{\nu+2n}_{-2n}(q)\operatorname{% me}_{\nu+2n}\left(z,q\right),$
##### 5: 28.4 Fourier Series
28.4.9 $2\left(A^{2n}_{0}(q)\right)^{2}+\sum_{m=1}^{\infty}\left(A^{2n}_{2m}(q)\right)% ^{2}=1,$
28.4.12 $\sum_{m=0}^{\infty}\left(B^{2n+2}_{2m+2}(q)\right)^{2}=1.$
##### 6: 28.28 Integrals, Integral Representations, and Integral Equations
28.28.20 $\dfrac{2}{\pi}\int_{0}^{\pi}\mathcal{C}^{(j)}_{2\ell}(2hR)\cos\left(2\ell\phi% \right)\operatorname{ce}_{2m}\left(t,h^{2}\right)\,\mathrm{d}t=\varepsilon_{% \ell}(-1)^{\ell+m}A^{2m}_{2\ell}(h^{2}){\operatorname{Mc}^{(j)}_{2m}}\left(z,h% \right),$
28.28.36 $\dfrac{\sinh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\cos t\operatorname{se}_{n}\left% (t,h^{2}\right)\operatorname{se}_{m}\left(t,h^{2}\right)}{{\sinh}^{2}z+{\sin}^% {2}t}\,\mathrm{d}t=(-1)^{p+1}\mathrm{i}h\widehat{\alpha}_{n,m}^{(s)}% \operatorname{Ds}_{0}\left(n,m,z\right),$
28.28.37 $\dfrac{\cosh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin t\operatorname{se}_{n}'% \left(t,h^{2}\right)\operatorname{se}_{m}\left(t,h^{2}\right)}{{\sinh}^{2}z+{% \sin}^{2}t}\,\mathrm{d}t=(-1)^{p+1}\mathrm{i}h\widehat{\alpha}_{n,m}^{(s)}% \operatorname{Ds}_{1}\left(n,m,z\right),$
28.28.38 $\widehat{\alpha}_{n,m}^{(s)}=\dfrac{1}{2\pi}\int_{0}^{2\pi}\cos t\operatorname% {se}_{n}\left(t,h^{2}\right)\operatorname{se}_{m}\left(t,h^{2}\right)\,\mathrm% {d}t=(-1)^{p}\dfrac{2}{\mathrm{i}\pi}\dfrac{\operatorname{se}_{n}'\left(0,h^{2% }\right)\operatorname{se}_{m}'\left(0,h^{2}\right)}{h\operatorname{Ds}_{2}% \left(n,m,0\right)}.$
28.28.41 $\dfrac{\cosh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin t\operatorname{se}_{n}\left% (t,h^{2}\right)\operatorname{ce}_{m}\left(t,h^{2}\right)}{{\sinh}^{2}z+{\sin}^% {2}t}\,\mathrm{d}t=(-1)^{p+1}\mathrm{i}h\widehat{\beta}_{n,m}\operatorname{Dsc% }_{0}\left(n,m,z\right),$
##### 7: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
28.24.2 $\varepsilon_{s}{\operatorname{Mc}^{(j)}_{2m}}\left(z,h\right)=(-1)^{m}\sum_{% \ell=0}^{\infty}(-1)^{\ell}\frac{A_{2\ell}^{2m}(h^{2})}{A_{2s}^{2m}(h^{2})}% \left(J_{\ell-s}\left(he^{-z}\right){\cal C}_{\ell+s}^{(j)}(he^{z})+J_{\ell+s}% \left(he^{-z}\right){\cal C}_{\ell-s}^{(j)}(he^{z})\right),$
28.24.3 ${\operatorname{Mc}^{(j)}_{2m+1}}\left(z,h\right)=(-1)^{m}\sum_{\ell=0}^{\infty% }(-1)^{\ell}\frac{A_{2\ell+1}^{2m+1}(h^{2})}{A_{2s+1}^{2m+1}(h^{2})}\left(J_{% \ell-s}\left(he^{-z}\right){\cal C}_{\ell+s+1}^{(j)}(he^{z})+J_{\ell+s+1}\left% (he^{-z}\right){\cal C}_{\ell-s}^{(j)}(he^{z})\right),$
28.24.4 ${\operatorname{Ms}^{(j)}_{2m+1}}\left(z,h\right)=(-1)^{m}\sum_{\ell=0}^{\infty% }(-1)^{\ell}\frac{B_{2\ell+1}^{2m+1}(h^{2})}{B_{2s+1}^{2m+1}(h^{2})}\left(J_{% \ell-s}\left(he^{-z}\right){\cal C}_{\ell+s+1}^{(j)}(he^{z})-J_{\ell+s+1}\left% (he^{-z}\right){\cal C}_{\ell-s}^{(j)}(he^{z})\right),$
28.24.5 ${\operatorname{Ms}^{(j)}_{2m+2}}\left(z,h\right)=(-1)^{m}\sum_{\ell=0}^{\infty% }(-1)^{\ell}\frac{B_{2\ell+2}^{2m+2}(h^{2})}{B_{2s+2}^{2m+2}(h^{2})}\left(J_{% \ell-s}\left(he^{-z}\right){\cal C}_{\ell+s+2}^{(j)}(he^{z})-J_{\ell+s+2}\left% (he^{-z}\right){\cal C}_{\ell-s}^{(j)}(he^{z})\right),$
For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).
##### 9: 31.14 General Fuchsian Equation
31.14.1 ${\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\sum_{j=1}^{N}\frac{\gamma_% {j}}{z-a_{j}}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\left(\sum_{j=1}^{N}\frac{% q_{j}}{z-a_{j}}\right)w=0},$ $\sum_{j=1}^{N}q_{j}=0$.
31.14.3 $w(z)=\left(\prod_{j=1}^{N}(z-a_{j})^{-\gamma_{j}/2}\right)W(z),$
##### 10: 28.22 Connection Formulas
28.22.1 ${\operatorname{Mc}^{(1)}_{m}}\left(z,h\right)=\sqrt{\dfrac{2}{\pi}}\dfrac{1}{g% _{\mathit{e},m}(h)\operatorname{ce}_{m}\left(0,h^{2}\right)}\operatorname{Ce}_% {m}\left(z,h^{2}\right),$
28.22.2 ${\operatorname{Ms}^{(1)}_{m}}\left(z,h\right)=\sqrt{\dfrac{2}{\pi}}\frac{1}{g_% {\mathit{o},m}(h)\operatorname{se}_{m}'\left(0,h^{2}\right)}\operatorname{Se}_% {m}\left(z,h^{2}\right),$
28.22.5 $g_{\mathit{e},2m}(h)=(-1)^{m}\sqrt{\dfrac{2}{\pi}}\dfrac{\operatorname{ce}_{2m% }\left(\frac{1}{2}\pi,h^{2}\right)}{A_{0}^{2m}(h^{2})},$
28.22.6 $g_{\mathit{e},2m+1}(h)=(-1)^{m+1}\sqrt{\frac{2}{\pi}}\dfrac{\operatorname{ce}_% {2m+1}'\left(\frac{1}{2}\pi,h^{2}\right)}{hA_{1}^{2m+1}(h^{2})},$
28.22.7 $g_{\mathit{o},2m+1}(h)=(-1)^{m}\sqrt{\dfrac{2}{\pi}}\dfrac{\operatorname{se}_{% 2m+1}\left(\frac{1}{2}\pi,h^{2}\right)}{hB_{1}^{2m+1}(h^{2})},$