sn(u,k) などの Lambert Series まとめ

文献

・ Evaluations of infinite series involving reciprocal hyperbolic functions ／Ce Xu
・ Hyperbolic summations derived using the Jacobi functions dc and nc ／John M. Campbell
・ INFINITE FAMILIES OF EXACT SUMS OF SQUARES FORMULAS, JACOBI ELLIPTIC FUNCTIONS, CONTINUED FRACTIONS, AND SCHUR FUNCTIONS ／Stephen C. Milne
・ The best-known properties and formulas for Jacobi functions(wolfram.com)

$\begin{array}{rl}{\displaystyle \mathrm{s}\mathrm{n}(K(x)z,x)}& =\frac{\pi }{xK(x)}\sum _{n=0}^{\mathrm{\infty }}\frac{\sin \frac{\pi (2n+1)}{2}z}{\sinh \frac{\pi (2n+1)}{2}\frac{K^{\prime }(x)}{K(x)}}\\ & ={(K(x)z)}^1-\frac{1+x^2}{3!}{(K(x)z)}^3+\frac{1+14x^2+x^4}{5!}{(K(x)z)}^5-\frac{(1+x^2)(1+134x^2+x^4)}{7!}{(K(x)z)}^7+\frac{1+1228x^2+5478x^4+1228x^6+x^8}{9!}{(K(x)z)}^9-\cdots \\ \mathrm{c}\mathrm{n}(K(x)z,x)& =\frac{\pi }{xK(x)}\sum _{n=0}^{\mathrm{\infty }}\frac{\cos \frac{\pi (2n+1)}{2}z}{\cosh \frac{\pi (2n+1)}{2}\frac{K^{\prime }(x)}{K(x)}}\\ & =1-\frac{1}{2!}{(K(x)z)}^2+\frac{1+4x^2}{4!}{(K(x)z)}^4-\frac{1+44x^2+16x^4}{6!}{(K(x)z)}^6+\frac{1+408x^2+912x^4+64x^6}{8!}{(K(x)z)}^8-\cdots \\ \mathrm{d}\mathrm{n}(K(x)z,x)& =\frac{\pi }{2K(x)}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\cos \pi nz}{\cosh \pi n\frac{K^{\prime }(x)}{K(x)}}\\ & =1-\frac{x^2}{2!}{(K(x)z)}^2+\frac{x^2(4+x^2)}{4!}{(K(x)z)}^4-\frac{x^2(16+44x^2+x^4)}{6!}{(K(x)z)}^6+\frac{x^2(64+912x^2+408x^4+x^6)}{8!}{(K(x)z)}^8-\cdots \\ \mathrm{n}\mathrm{s}(K(x)z,x)& =\frac{\pi }{2K(x)}(\frac{1}{\sin \frac{\pi }{2}z}+4\sum _{n=0}^{\mathrm{\infty }}\frac{\sin \frac{\pi (2n+1)}{2}z}{e^{\pi (2n+1)\frac{K^{\prime }(x)}{K(x)}}-1})\\ & =\frac{1}{K(x)z}+\frac{1+x^2}{6}{(K(x)z)}^1+\frac{7-22x^2+7x^4}{360}{(K(x)z)}^3+\frac{(1+x^2)(31-46x^2+31x^4)}{15120}{(K(x)z)}^5+\frac{127-284x^2+186x^4-284x^6+127x^8}{604800}{(K(x)z)}^7+\cdots \\ \mathrm{n}\mathrm{c}(K(x)z,x)& =\frac{\pi }{2\sqrt{1-x^2}K(x)}(\frac{1}{\cos \frac{\pi }{2}z}-4\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n\cos \frac{\pi (2n+1)}{2}z}{e^{\pi (2n+1)\frac{K^{\prime }(x)}{K(x)}}+1})\\ & =1+\frac{1}{2!}(K(x)z{)}^2+\frac{5-4x^2}{4!}(K(x)z{)}^4+\frac{61-76x^2+16x^4}{6!}(K(x)z{)}^6+\frac{1385-2424x^2+1104x^4-64x^6}{8!}(K(x)z{)}^8+\cdots \\ \mathrm{n}\mathrm{d}(K(x)z,x)& =\frac{\pi }{2\sqrt{1-x^2}K(x)}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^n\cos \pi nz}{\cosh \pi n\frac{K^{\prime }(x)}{K(x)}}\\ & =1+\frac{x^2}{2!}{(K(x)z)}^2+\frac{x^2(5x^2-4)}{4!}{(K(x)z)}^4+\frac{x^2(61x^2-76x^2+16)}{6!}{(K(x)z)}^6+\frac{x^2(1385x^6-2424x^4+1104x^2-64)}{8!}{(K(x)z)}^8+\cdots \\ \mathrm{s}\mathrm{d}(K(x)z,x)& =\frac{2\pi }{x\sqrt{1-x^2}K(x)}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n\sin \frac{\pi (2n+1)}{2}z}{\cosh \frac{\pi (2n+1)}{2}\frac{K^{\prime }(x)}{K(x)}}\\ & ={(K(x)z)}^1+\frac{2x^2-1}{3!}{(K(x)z)}^3+\frac{16x^4-16x^2+1}{5!}{(K(x)z)}^5+\frac{(2x^2-1)(136x^4-136x^2+1)}{7!}{(K(x)z)}^7+\cdots \\ \mathrm{c}\mathrm{d}(K(x)z,x)& =\frac{\pi }{xK(x)}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n\cos \frac{\pi (2n+1)}{2}z}{\sinh \frac{\pi (2n+1)}{2}\frac{K^{\prime }(x)}{K(x)}}\\ & =1-\frac{1-x^2}{2!}(K(x)z{)}^2+\frac{(1-x^2)(1-5x^2)}{4!}(K(x)z{)}^4-\frac{(1-x^2)(1-46x^2+61x^4)}{6!}(K(x)z{)}^6+\frac{(1-x^2)(1-411x^2+1731x^4-1385x^6)}{8!}(K(x)z{)}^8-\cdots \\ \mathrm{s}\mathrm{c}(K(x)z,x)& =\frac{\pi }{2\sqrt{1-x^2}K(x)}(\tan \frac{\pi }{2}z+4\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n\sin \pi nz}{e^{2\pi n\frac{K^{\prime }(x)}{K(x)}}-1})\\ & =(K(x)z{)}^1+\frac{2-x^2}{3!}{(K(x)z)}^3+\frac{16-16x^2+x^4}{5!}{(K(x)z)}^5+\frac{(2-x^2)(136-136x^2+x^4)}{7!}{(K(x)z)}^7+\frac{7936-15872x^2+9168x^4-1232x^6+x^8}{9!}{(K(x)z)}^9+\cdots \\ \mathrm{d}\mathrm{c}(K(x)z,x)& =\frac{\pi }{2K(x)}(\frac{1}{\cos \frac{\pi }{2}z}+4\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n\cos \frac{\pi (2n+1)}{2}z}{e^{\pi (2n+1)\frac{K^{\prime }(x)}{K(x)}}-1})\\ & =1+\frac{1-x^2}{2!}(K(x)z{)}^2+\frac{(1-x^2)(5-x^2)}{4!}(K(x)z{)}^4+\frac{(1-x^2)(61-46x^2+x^4)}{6!}(K(x)z{)}^6+\frac{(1-x^2)(1385-1731x^2+411x^4-x^6)}{8!}(K(x)z{)}^8+\cdots \end{array}$

$\begin{array}{rl}{\displaystyle {\mathrm{s}\mathrm{n}}^2(K(x)z,x)}& =\frac{1}{x^2}(1-\frac{E(x)}{K(x)})-\frac{{\pi }^2}{x^{2}K(x{)}^2}\sum _{n=1}^{\mathrm{\infty }}\frac{n\cos \pi nz}{\sinh \pi n\frac{K^{\prime }(x)}{K(x)}}\\ & ={(K(x)z)}^2-\frac{1+x^2}{3}{(K(x)z)}^4+\frac{2+13x^2+2x^4}{45}{(K(x)z)}^6-\frac{(1+x^2)(1+29x^2+x^4)}{315}{(K(x)z)}^8+\cdots \\ {\mathrm{n}\mathrm{s}}^2(K(x)z,x)& =1-\frac{E(x)}{K(x)}+\frac{1}{{\sin }^2\frac{\pi }{2}z}-\frac{2{\pi }^2}{K(x{)}^2}\sum _{n=1}^{\mathrm{\infty }}\frac{n\cos \pi nz}{e^{\pi n\frac{K^{\prime }(x)}{K(x)}}-1}\\ & =\frac{1}{(K(x)z{)}^2}+\frac{1+x^2}{3}+\frac{1-x^2+x^4}{15}(K(x)z{)}^2+\frac{(1+x^2)(1-2x^2)(2-x^2)}{189}(K(x)z{)}^4+\frac{(1-x^2+x^4{)}^2}{675}(K(x)z{)}^6+\cdots \\ {\mathrm{s}\mathrm{c}}^2(K(x)z,x)& =\frac{1}{1-x^2}(-\frac{E(x)}{K(x)}+\frac{{\pi }^2}{4K(x{)}^2}\frac{1}{{\cos }^2\frac{\pi }{2}z}+\frac{2{\pi }^2}{K(x{)}^2}\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n-1}n\cos \pi nz}{e^{2\pi n\frac{K^{\prime }(x)}{K(x)}}-1})\\ & ={(K(x)z)}^2+\frac{2-x^2}{3}{(K(x)z)}^4+\frac{17-17x^2+2x^4}{45}{(K(x)z)}^6+\frac{(2-x^2)(31-31x^2+x^4)}{315}{(K(x)z)}^8+\frac{1382-2764x^2+1641x^4-259x^6+2x^8}{14175}{(K(x)z)}^{10}+\cdots \\ {\mathrm{s}\mathrm{d}}^2(K(x)z,x)& =\frac{1}{x^2(1-x^2)}(\frac{E(x)}{K(x)}-(1-x^2)+\frac{{\pi }^2}{K(x{)}^2}\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n-1}n\cos \pi nz}{\sinh \pi n\frac{K^{\prime }(x)}{K(x)}})\\ & ={(K(x)z)}^2+\frac{2x^2-1}{3}{(K(x)z)}^4+\frac{17x^4-17x^2+2}{45}{(K(x)z)}^6+\frac{(2x^2-1)(31x^4-31x^2+1)}{315}{(K(x)z)}^8+\frac{1382x^8-2764x^6+1641x^4-259x^2+2}{14175}{(K(x)z)}^{10}+\cdots \end{array}$

$\begin{array}{rl}{\displaystyle \frac{\mathrm{s}\mathrm{n}(K(x)z,x)\mathrm{c}\mathrm{n}(K(x)z,x)}{\mathrm{d}\mathrm{n}(K(x)z,x)}}& =\frac{2\pi }{x^{2}K(x)}\sum _{n=0}^{\mathrm{\infty }}\frac{\sin \pi (2n+1)z}{\sinh \pi (2n+1)\frac{K^{\prime }(x)}{K(x)}}\\ & ={(K(x)z)}^1+\frac{x^2-2}{3}{(K(x)z)}^3+\frac{2(x^4-x^2+1)}{15}{(K(x)z)}^5+\frac{(x^2-2)(17x^4-2x^2+2)}{315}{(K(x)z)}^7+\cdots \\ \frac{\mathrm{s}\mathrm{n}(K(x)z,x)\mathrm{d}\mathrm{n}(K(x)z,x)}{\mathrm{c}\mathrm{n}(K(x)z,x)}& =\frac{\pi }{2K(x)}(\tan \frac{\pi }{2}z+4\sum _{n=1}^{\mathrm{\infty }}\frac{\sin \pi nz}{e^{\pi n\frac{K^{\prime }(x)}{K(x)}}+{(-1)}^n})\\ & ={(K(x)z)}^1+\frac{1-2x^2}{3}{(K(x)z)}^3+\frac{2(1-x^2+x^4)}{15}{(K(x)z)}^5+\frac{(1-2x^2)(17-2x^2+2x^4)}{315}{(K(x)z)}^7+\cdots \\ \frac{\mathrm{s}\mathrm{n}(K(x)z,x)}{\mathrm{c}\mathrm{n}(K(x)z,x)\mathrm{d}\mathrm{n}(K(x)z,x)}& =\frac{\pi }{2(1-x^2)K(x)}(\tan \frac{\pi }{2}z+4\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n\sin \pi nz}{e^{\pi n\frac{K^{\prime }(x)}{K(x)}}+1})\\ & ={(K(x)z)}^1+\frac{1+x^2}{3}{(K(x)z)}^3+\frac{2(1-x^2+x^4)}{15}{(K(x)z)}^5+\frac{(1+x^2)(17-32x^2+17x^4)}{315}{(K(x)z)}^7+\cdots \\ \frac{{\mathrm{s}\mathrm{n}}^2(K(x)z,x){\mathrm{c}\mathrm{n}}^2(K(x)z,x)}{{\mathrm{d}\mathrm{n}}^2(K(x)z,x)}& =\frac{1}{x^4}(2-x^2-\frac{2E(x)}{K(x)}-\frac{4{\pi }^2}{K(x{)}^2}\sum _{n=1}^{\mathrm{\infty }}\frac{n\cos 2\pi nz}{\sinh 2\pi n\frac{K^{\prime }(x)}{K(x)}})\\ & ={(K(x)z)}^2+\frac{2(x^2-2)}{3}{(K(x)z)}^4+\frac{17x^4-32x^2+32}{45}{(K(x)z)}^6+\frac{2(x^2-2)(31x^4-16x^2+16)}{315}{(K(x)z)}^8+\cdots \\ \frac{{\mathrm{s}\mathrm{n}}^2(K(x)z,x){\mathrm{d}\mathrm{n}}^2(K(x)z,x)}{{\mathrm{c}\mathrm{n}}^2(K(x)z,x)}& =1-\frac{2E(x)}{K(x)}+\frac{{\pi }^2}{4K(x{)}^2}\frac{1}{{\cos }^2\frac{\pi }{2}z}-\frac{2{\pi }^2}{K(x{)}^2}\sum _{n=1}^{\mathrm{\infty }}\frac{n\cos \pi nz}{e^{\pi n\frac{K^{\prime }(x)}{K(x)}}-{(-1)}^n}\\ & ={(K(x)z)}^2+\frac{2(1-2x^2)}{3}{(K(x)z)}^4+\frac{17-32x^2+32x^4}{45}{(K(x)z)}^6+\frac{(1-2x^2)(31-16x^2+16x^4)}{315}{(K(x)z)}^8+\cdots \end{array}$