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幟äœçŽæ°$\D{_2}F_1\L[\BM s,1-s\\1\EM;x^2\R]$ã®$\text{Fourier-Legendre~Expansion}$ãèšç®ããŸããïŒçµæã¯æ¬¡ã®éãã§ãïŒ
å®çïŒ$s$ã¯$|s|<1$ãæºããæ°ã§ïŒ$r$ãæŽæ°ãšããïŒ$P_n(x)$ã¯$n$次ã®$\rm Legendre$å€é
åŒãšããïŒãã®ãšã以äžãæãç«ã€ïŒ
ãã ãïŒ$\D\gamma_n=\frac{{\L(\frac{1}{2},s,1-s\R)}_n}{{\L(1,\frac{1}{2}+s,\frac{3}{2}-s\R)}_n}$ãšããŠããŸãïŒ${_2}F_1[\cdot]$ã¯è¶ 幟äœçŽæ°ã§ïŒ$\D{_2}F_1\L[\BM a,b\\c\EM;x\R]=\sum_{n=0}^\infty \frac{{(a,b)}_n}{{(c)}_nn!}\,x^n$ã§ãïŒ$\D{(z)}_n=\prod_{k=1}^n(k-1+z),\quad (a,b)_n=(a)_n(b)_n$ã§ãïŒ
ã颿°$f(x)$ã$\text{Legendre}$å€é åŒ$P_n(x)$ã®åã§è¡šãããšã$\text{Fourier-Legendre~Expansion}$(ç¥ããŠ$\rm FL$å±é)ãšåŒã³ãŸãïŒä»¥äžã«$P_n(x)$ã®åºæ¬çãªæ§è³ªããŸãšããŸãïŒ
å®çŸ© | $\BA\D P_n(x)=\sum_{k=0}^n \binom{n}{k}\binom{k+n}{k}\L(\frac{x-1}{2}\R)^n\EA$ |
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çŽäº€æ§ | $\D\int_{-1}^1 P_m(x)P_n(x)\,dx=\begin{cases}\dfrac{2}{2n+1} &(m=n)\\ \,\,\,\,\,\,\,~0 &(m\neq n)\end{cases}$ |
$\textrm{FL}$å±éä¿æ° | $\BA\D f_n=\frac{2n+1}{2}\int_{-1}^1f(x)P_n(x)\,dx \EA$ |
$\textrm{FL}$å±é | $\BA\D f(x)=\sum_{n=0}^\infty f_nP_n(x)\EA$ |
å
ç© | $\BA\D \langle f(x),g(x)\rangle=\int_{-1}^1 f(x)g(x)\,dx=\sum_{n=0}^\infty \frac{2f_ng_n}{2n+1} \EA$ |
挞ååŒïŒ | $\BA\D (n+1)P_{n+1}(x)-(2n+1)xP_n(x)+nP_{n-1}(x)=0\EA$ |
挞ååŒïŒ | $\BA\D P_{n+1}'(x)=(n+1)P_n(x)+xP_n'(x)\EA$ |
挞ååŒïŒ | $\BA\D (2n+1)P_n(x)=P_{n+1}'(x)-P_{n-1}'(x)\EA$ |
挞ååŒïŒ | $\BA\D P_{n}'(x)=\frac{n(n+1)}{2n+1}\frac{P_{n-1}(x)-P_{n+1}(x)}{1-x^2}\EA$ |
ã詳现ã¯äŸãã° ${\color{gray}{\rm Wolfram}}{\color{teal}{\rm MathW{\tiny\!\! ð}rld}}$ ã $\rm \color{black}{Wikipedia},$ ${\color{black}{\rm H{\small AND}}}{\color{orange}{\rm W}}{\color{black}{\small\rm IKI}}$ ã«èšèŒãããŠããŸãã${\rm arXiv}$ã§æ€çŽ¢ããã°ïŒ $\rm Lecture~notes~on~Legendre~polynomials:~their~origin~and~main~properties$ ${(\rm FÂŽabio M. S. Lima)}$ãªã©ããããŸãã
ããã®$\rm FL$å±éã§ããïŒããèªäœãäž»é¡ãšããæç®ã¯æ€çŽ¢ããŠãããŸãèŠã€ãããŸããã§ããïŒå®éã®æç®ãããã€ã瀺ããŸãïŒ
$\qquad\surd$
$\rm{Apâery}\textrm{-Type~Series~and~Colored~Multiple~Zeta~Values}$
$({\rm Ce~Xua, ~Jianqiang~Zhao})$
$\qquad\surd$
$\textrm{On~the~interplay~between~hypergeometric~series,~Fourier-Legendre~expansions~and~Euler~sums}$
$({\rm Marco~Cantarini,~Jacopo~DâAurizio})$
$\qquad\surd$
$\textrm{A~NOTE~ON~CLEBSCH-GORDAN~INTEGRAL,~FOURIER-LEGENDRE~EXPANSIONS~AND~CLOSED~FORM~FOR~HYPERGEOMETRIC~SERIES}$
$({\rm Marco~Cantarini})$
$\qquad\surd$
$\rm New~families~of~double~hypergeometric~series~for~constants~involving~\frac{1}{\pi^2}$
$({\rm John~Campbell})$
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ã§ãããããèŠãŠïŒ$K(x)$ã®$\rm FL$å±éã¯ã©ããªãã®ããšæããŸãããïŒæ€çŽ¢ããŠãèŠã€ãããŸããã§ããã®ã§ïŒèªåã§èããŸããïŒ
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ã$K(x)$ã®$\rm FL$å±éä¿æ°ã®åŒã¯ïŒæŒžååŒãå°åºãæšæž¬ããåŒããããæºããããšã§èšŒæããŸããïŒãã®éã«è£å©çã«æçšã§ãã£ãã®ã第äºçš®å®å šæ¥åç©å$E(x)$ã§ãïŒ$K(x)$ãš$E(x)$ã®ããã ã«ã¯ïŒåŸ®åæ¹çšåŒã$\rm Legendre~relation$ãšãã£ãéœåãããé¢ä¿åŒãæãç«ã¡ãŸãïŒãã®åŸïŒæ¬¡ã®æç®ã®äžã«è峿·±ãåŒãèŠã€ããŸããïŒ
$\qquad\surd$ $\textrm{Legendre-type relations for generalized complete elliptic integrals}$ $(\rm J. G. WAN )$
ããã ãïŒ$\D K_s(x)=\frac{\pi}{2}{_2}F_1\L[\BM\frac{1}{2}-s,\frac{1}{2}+s\\1\EM;x^2\R],\quad E_s(x)=\frac{\pi}{2}{_2}F_1\L[\BM-\frac{1}{2}-s,\frac{1}{2}+s\\1\EM;x^2\R],\quad K_s'(x)=K_s(\sqrt{1-x^2}),\quad E_s'(x)=E_s(\sqrt{1-x^2})$ã§ãïŒ
ãããåããŠïŒ$\D\frac{\pi}{2}{_2}F_1\L[\BM s,1-s\\1\EM;x^2\R]$ã®$\rm FL$å±éä¿æ°ãèšç®ã§ããã®ã§ã¯ãªãããšæããŸããïŒçµæãšããŠæ¬¡ã®éãã«èšç®ã§ããŸããïŒ
å®çïŒ$s$ã¯$|s|<1$ãæºããæ°ã§ïŒ$r$ãæŽæ°ãšããïŒ$P_n(x)$ã¯$n$次ã®$\rm Legendre$å€é åŒãšããïŒãã®ãšã以äžãæãç«ã€ïŒ
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å®çŸ©ïŒ | $\BA\D K_s=K_s(x)=\frac{\pi}{2}{_2}F_1\L[\BM s,1-s\\1\EM;x^2\R]\EA$ |
---|---|
å®çŸ©ïŒ | $\BA\D E_s=E_s(x)=\frac{\pi}{2}{_2}F_1\L[\BM s,-s\\1\EM;x^2\R]\EA$ |
å®çŸ©ïŒ | $\BA\D P_n=P_n(x)\EA$ |
å®çŸ©ïŒ | $\BA\D K_n=\int_0^1 K_sP_{2n}\,dx\EA$ |
å®çŸ©ïŒ | $\BA\D E_n=\int_0^1 E_sP_{2n}\,dx\EA$ |
åŸ®åæ¹çšåŒïŒ | $\BA\D \frac{d}{dx}K_s=\frac{2s}{x}\L(\frac{E_s}{1-x^2}-K_s\R)\EA$ |
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åŸ®åæ¹çšåŒïŒ | $\BA\D \frac{d}{dx}E_s=\frac{2s}{x}(E_s-K_s)\EA$ |
åŸ®åæ¹çšåŒïŒ | $\BA\D \frac{d}{dx}xK_s=\frac{2sE_s}{1-x^2}+(1-2s)K_s\EA$ |
åŸ®åæ¹çšåŒïŒ | $\BA\D \frac{d}{dx}xE_s=(1+2s)E_s-2sK_s\EA$ |
挞ååŒïŒ | $\BA\D P_{n+1}'(x)=(n+1)P_n(x)+xP_n'(x)\EA$ |
挞ååŒïŒ | $\BA\D (2n+1)P_n(x)=P_{n+1}'(x)-P_{n-1}'(x)\EA$ |
挞ååŒïŒ | $\BA\D P_{n}'(x)=\frac{n(n+1)}{2n+1}\frac{P_{n-1}(x)-P_{n+1}(x)}{1-x^2}\EA$ |
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