$$\newcommand{BA}[0]{\begin{align*}}
\newcommand{BE}[0]{\begin{equation}}
\newcommand{bl}[0]{\boldsymbol}
\newcommand{D}[0]{\displaystyle}
\newcommand{EA}[0]{\end{align*}}
\newcommand{EE}[0]{\end{equation}}
\newcommand{h}[0]{\boldsymbol{h}}
\newcommand{k}[0]{\boldsymbol{k}}
\newcommand{L}[0]{\left}
\newcommand{l}[0]{\boldsymbol{l}}
\newcommand{m}[0]{\boldsymbol{m}}
\newcommand{n}[0]{\boldsymbol{n}}
\newcommand{R}[0]{\right}
$$
$\Large ๐ธ๐๐๐๐๐๐๐๐๐๐๐$
ใ
ใใฎ่จไบ
ใง่ฟฐในใใใใซ๏ผ$\textrm{WZ-pair}$ใฏ
\begin{align*}
F(i,j)+G(i+1,j)=G(i,j)+F(i,j+1)
\end{align*}
ใๆบใใ$F,G$ใฎ็ตใงใใใใใใฏ่ฆ่ฆ็ใซ่กจใใจ๏ผๅนณ้ขไธใฎๆ ผๅญใฐใฉใใซใใใฆ
\begin{align*}
&F(i,j):{\rm weight~on~}(i,j)\to(i+1,j)\\
&G(i,j):{\rm weght~on~}(i,j)\to(i,j+1)
\end{align*}
ใจใใใจใใซ๏ผๅง็น$(a,b)$ใใ็ต็น$(a',b')$ใพใงใซ้ใๆ ผๅญใฎ้ใฟใฎๅใ๏ผใฉใฎใใใช็ต่ทฏใใใฉใฃใฆใไธ่ดใใใจใใใใจใงใใใ
${\rm path~invariance~in~multivariable}$
ใไธ่ฌใซ$n$ๅใฎใใฉใกใผใฟใซใใฃใฆ${\rm WZ~}{n\textrm{-tuple}}$ใ
${\bf Definition.}\quad (x_1,\cdots,x_i,\cdots,x_n)$ใใ$(x_1,\cdots,x_i+1,\cdots,x_n)$ใธใฎ้ใฟใ$F_i(x_1,\cdots,x_n)$ใจใ๏ผ$j\neq k$ใซๅฏพใใฆ$F_j(x_1,\cdots,x_n)$ใจ$F_k(x_1,\cdots,x_n)$ใ$\textrm{WZ-pair}$ใงใใใจใ๏ผ
$\BA\D\\
\big(F_1(x_1,\cdots,x_n),\cdots,F_n(x_1,\cdots,x_n)\big)
\EA$
ใ${\rm WZ~}{n\textrm{-tuple}}$ใจๅผใถใ
ใจๅฎ็พฉใใพใใใใใซใใ๏ผ${\mathbb R}^n$ไธใง$\rm path~invariant~grid~graph$ใๅฝขๆใใใพใใ
${\rm WZ~}n\textrm{-tuple}$ใฎๆงๆ
ใใปใจใใฉใฎ${\rm WZ~}n\textrm{-tuple}$ใฏ๏ผๆฌกใฎๅใคใฎๅ
ธๅ็ใช่ถ
ๅนพไฝ็ดๆฐใฎๅ
ฌๅผใจ้ข้ฃใใฆใใใจ่จใใใฆใใพใใ
$\qquad\textcolor{red}โ$$\rm the~binomial~theorem$
$\qquad\textcolor{red}โ$$\rm Gauss{'}s~theorem$
$\qquad\textcolor{red}โ$${\rm the~}\textrm{Pfaff-}{\rm Saalsch\ddot{u}tz~theorem}$
$\qquad\textcolor{red}โ$${\rm Dougall{'}s~}\textrm{very-well-poised}~{_7F_6}~{\rm summation~theorem}$
${\rm WZ~}\textrm{4-tuple}$ใฎๆงๆ
$\quad{\rm WZ~}\textrm{4-tuple}$ใฏ$\rm Gauss{'}s~theorem$ใจ้ขไฟใใฆใใพใใ$\rm Gauss{'}s~theorem$ใจใฏ๏ผ
$\BA\D\\
\sum_{n=0}^\infty \frac{(a)_n(b)_n}{(c)_nn!}=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}
\EA$
ใงใใใใใๆธใๆใใใจ๏ผ
$\BA\D\\
\sum_{n=0}^\infty \frac{\Gamma(a+n)\Gamma(b+n)\Gamma(a+c)\Gamma(b+c)}{\Gamma(a)\Gamma(b)\Gamma(c)\Gamma(n+1)\Gamma(a+b+c+n)}=1
\EA$
ใจใใใใใกใซใชใใพใใใใใใใจใซ่ใใใจ๏ผ
$\BA\D\\
H(x_1,x_2,x_3,x_4)=\frac{\Gamma(a_1+a_3+x_1+x_3)\Gamma(a_1+a_4+x_1+x_4)\Gamma(a_2+a_3+x_2+x_3)\Gamma(a_2+a_4+x_2+x_4)}{\Gamma(a_1+x_1)\Gamma(a_2+x_2)\Gamma(a_3+x_3)\Gamma(a_4+x_4)\Gamma(a_1+a_2+a_3+a_4+x_1+x_2+x_3+x_4)}
\EA$
ใจใ๏ผๅ
ใปใฉใฎๅฎ็พฉใซใใใฆ
$\BA\D\\
&F_i(x_1,x_2,x_3,x_4)=\frac{H(x_1,x_2,x_3,x_4)}{a_i+x_i}\qquad\quad (i=1,2)\\
&F_i(x_1,x_2,x_3,x_4)=-\frac{H(x_1,x_2,x_3,x_4)}{a_i+x_i}\qquad~ (i=3,4)
\EA$
ใจใใ่กจ็คบใๆใ็ซใคใใจใใใใใพใใใ
ๅ
ทไฝ็ใซๆธใใจ
$\BA\D\\
F_1(x_1,x_2,x_3,x_4)-F_1(x_1,x_2+1,x_3,x_4)&=F_2(x_1,x_2,x_3,x_4)-F_2(x_1+1,x_2,x_3,x_4)\\
F_1(x_1,x_2,x_3,x_4)-F_1(x_1,x_2,x_3+1,x_4)&=F_3(x_1,x_2,x_3,x_4)-F_3(x_1+1,x_2,x_3,x_4)\\
F_1(x_1,x_2,x_3,x_4)-F_1(x_1,x_2,x_3,x_4+1)&=F_4(x_1,x_2,x_3,x_4)-F_4(x_1+1,x_2,x_3,x_4)\\
F_2(x_1,x_2,x_3,x_4)-F_2(x_1,x_2,x_3+1,x_4)&=F_3(x_1,x_2,x_3,x_4)-F_3(x_1,x_2+1,x_3,x_4)\\
F_2(x_1,x_2,x_3,x_4)-F_2(x_1,x_2,x_3,x_4+1)&=F_4(x_1,x_2,x_3,x_4)-F_4(x_1,x_2+1,x_3,x_4)\\
F_3(x_1,x_2,x_3,x_4)-F_3(x_1,x_2,x_3,x_4+1)&=F_4(x_1,x_2,x_3,x_4)-F_4(x_1,x_2,x_3+1,x_4)
\EA$
ใๆใ็ซใกใพใใ
ใใพใ๏ผไธใคใฎ$\textrm{WZ-pair}$ใใใจใฅใใฆ๏ผๆฐใใช$\textrm{WZ-pair}$ใ็ๆใใใใใฎๅ็ดใชๆไฝใ๏ผใคๆใใใใพใใ
$\rm \textcolor{red}{Shifting}:$ใในใฆใฎใใฉใกใผใฟใฏ๏ผ็บๆฃใใชใ้ใ๏ผไปปๆใฎ่ค็ด ๆฐใซๅฏพใใฆๅนณ่ก็งปๅๅฏ่ฝใงใใใ
$\BA\D\\
\big(F_1(x_1,\cdots,x_n),\cdots,F_n(x_1,\cdots,x_n)\big)\to
\big(F_1(x_1+z_1,\cdots,x_n+z_n),\cdots,F_n(x_1+z_1,\cdots,x_n+z_n)\big)
\EA$
$\rm \textcolor{red}{Shadowing}:$ๅจๆใ$1$ใฎ้ขๆฐใๆใใใใฎใ${\rm WZ~}{n\textrm{-tuple}}$ใจใชใใ
$\BA\D\\
H(x_1,\cdots,x_n)\to \cos(\pi x_i)\Gamma(a_i+x_i)\Gamma(1-a_i-x_i)H(x_1,\cdots,x_n)
\EA$
$\rm \textcolor{red}{Symmetrization}:$$(F(x,y),G(x,y))$ใ$\textrm{WZ-pair}$ใงใใ$\BA\D\\
F(x,y)-F(x,y+1)=G(x,y)-G(x+1,y)
\EA$
$\qquad$ใๆใ็ซใคใจใ๏ผ$(G(-y,-x-1),F(-y-1,-x))$ใ$\textrm{WZ-pair}$ใจใชใใ$\BA\D\\
(F(x,y),G(x,y))\to(G(-y,-x-1),F(-y-1,-x))
\EA$
$\quad\rm Symmetrization$ใฎๅ
ๅฎนใฏๆๅไบๅฎใงใใ๏ผ็นๅฎใฎๅผ็งฐใใใใใฉใใใใใใชใใฃใฎใง๏ผใใใงใฏใใๅผใถใใจใซใใพใใใพใ๏ผ$(F,G)\neq (G,F)$ใงใใใใจใซๆณจๆใใฆใใ ใใใ
ใใใฎ๏ผใคใฎๆณๅใ็จใใใใจใง๏ผ๏ผใคใฎ${\rm WZ~}{n\textrm{-tuple}}$ใใใจใซๅฅใฎ${\rm WZ~}{n\textrm{-tuple}}$ใ็ๆใใใใจใใงใใพใใ
$\rm WZ~method~in~{\mathbb R}^4$
$\quad\mathbb R^4$ไธใฎ$\rm WZ~method$ใ่ใ๏ผ็ก้็ดๆฐใฎๆ็ญๅผใๅพใใใจใ็ฎๆใใพใใ
$\BA\D\\
H(i,j,k,l)=(-1)^k\frac{\Gamma(1-k)\Gamma(1+a+i+k)\Gamma(1-a+i+l)\Gamma(1+j+k)\Gamma(1-2a+j+l)}{\Gamma(1+a+i)\Gamma(1+j)\Gamma(-2a+l)\Gamma(2-a+i+j+k+l)}
\EA$
ใจใ๏ผ${\rm WZ~}{4\textrm{-tuple}}$ใ
$\BA\D\\
(F_1(i,j,k,l),F_2(i,j,k,l),F_3(i,j,k,l),F_4(i,j,k,l))
=\L(\frac{H(i,j,k,l)}{1+a+i},\frac{H(i,j,k,l)}{1+j},-\frac{H(i,j,k,l)}{k},-\frac{H(i,j,k,l)}{-2a+l}\R)
\EA$
ใจใใพใใใพใ๏ผไฝ็ฝฎ$L$ใใ$L'$ใพใงใฎ้ใฟใ$W[L\to L']$ใจๆธใใใจใซใใพใใ
ไฝ็ฝฎ$(0,0,0,0)$ใใ$(1,0,0,0)$ใใค้ฒใ็ต่ทฏใฎ้ใฟใฎๅใฏ
$\BA\D\\
&\sum_{n=0}^\infty W[(n,0,0,0)\to(n+1,0,0,0)]\\
=&\sum_{n=0}^\infty F_1(n,0,0,0)\\
=&\sum_{n=0}^\infty \frac{1}{1+a+n}\frac{\Gamma(1-a+n)\Gamma(1-2a)}{\Gamma(-2a)\Gamma(2-a+n)}\\
=&-2a\sum_{n=1}^\infty \frac{1}{n^2-a^2}
\EA$
ใจใชใใพใใ
ไฝ็ฝฎ$(0,0,0,0)$ใใ$(1,2,-1,-1)$ใใค้ฒใ็ต่ทฏใฎ้ใฟใฎๅใฏ
$\BA\D\\
\sum_{n=0}^\infty W[(n,2n,-n,-n)\to(n+1,2n+2,-n-1,-n-1)]
\EA$
ใจใชใใพใใใใใง๏ผ$W[(n,2n,-n,-n)\to(n+1,2n+2,-n-1,-n-1)]$ใฏ๏ผ$i\to j\to k\to l$ใฎ้ ใซ้ฒใใใฎใจใใฆ่จ็ฎใใใจ๏ผ
$\BA\D\\
&F_1(n,2n,-n,-n)+F_2(n+1,2n,-n,-n)+F_2(n+1,2n+1,-n,-n)\\
-&F_3(n+1,2n+2,-n-1,-n)-F_4(n+1,2n+2,-n-1,-n-1)
\EA$
ใซ็ญใใใชใใพใใใใใไบบๅใง่จ็ฎใใใฎใฏๆ้ใฎ็ก้งใงใใ๏ผ่จ็ฎ้ใ่จๅคงใงไบบใฎๆใงใใใใฎใใใชใใจๆใใใงใใใญใใชใใใใใใ่จ็ฎใใฆใใใใตใคใใฟใใใชใฎใใฃใใใใใงใใใญใใปใใจใใซใใใ็ฅใใพใใใใ
ใใใง๏ผ่จ็ฎใใ็ตๆใฏ
$\BA\D\\
-\frac{3a}{1-a^2}\sum_{n=0}^\infty \frac{(2)_n(1-2a,1+2a)_n}{2^{2n}\L(\frac{3}{2}\R)_n(2-a,2+a)_n}
\EA$
ใจใชใใใใใงใใใใใใซๆฐๅคใฏไธ่ดใใพใใใ
ใใใซ๏ผ$W[(n,0,0,0)\to(n,2n,-n,-n)]$ใฏ๏ผ$n\to\infty$ใง$0$ใซๅๆใ๏ผ
$\BA\D\\
-2a\sum_{n=1}^\infty \frac{1}{n^2-a^2}=-\frac{3a}{1-a^2}\sum_{n=0}^\infty \frac{(2)_n(1-2a,+2a)_n}{2^{2n}\L(\frac{3}{2}\R)_n(2-a,2+a)_n}
\EA$
ใใชใใก
$\BA\D\\
\sum_{n=1}^\infty \frac{1}{n^2-a^2}
=\sum_{n=1}^\infty \frac{3}{\binom{2n}{n}(n^2-a^2)}\prod_{m=1}^{n-1}\frac{m^2-4a^2}{m^2-a^2}
\EA$
ใจใใๅผใๅพใใใพใใ
ใใไธ่ฌใซใฏ๏ผ$H(x_1,x_2,x_3,x_4)=\cdots$ใฎๅผใซใใใฆ๏ผ$(a_1,a_2,a_3,a_4)=(1+a,1,0,-a-b)$ใจใใใใจใง
$\BA\D\\
\sum_{n=1}^\infty \frac{1}{(n-a)(n-b)}
=\sum_{n=1}^\infty \frac{3n-a-b}{n\binom{2n}{n}}\frac{(1-a+b,1+a-b)_{n-1}}{(1-a)_n(1-b)_n}
\EA$
ใจใชใใพใใ
ใใปใใซใๅ
ทไฝไพใ่จ็ฎใใฆใฟใพใใใ
$(0,0,0,0)$ใใ$(1,1,-1,0)$ใใค้ฒใใใจใง
$\BA\D\\
\sum_{n=1}^\infty \frac{1}{n^2-x^2}
=\sum_{n=1}^\infty \frac{(-1)^{n-1}(2n-x)}{n(n^2-x^2)}\frac{(1-2x)_{n-1}}{(1+x)_{n-1}}
\EA$
ใๅพใพใใ
$(0,0,0,0)$ใใ$(1,2,-1,0)$ใใค้ฒใใใจใง
$\BA\D\\
\sum_{n=1}^\infty \frac{1}{n^2-x^2}
=\sum_{n=1}^\infty \frac{(-1)^{n-1}(10n^2-3(1+2x)n+x)}{n\binom{2n}{n}}\frac{(1-x)_{n-1}(1-2x)_{2n-2}}{(1+x)_n(1-x)_{2n}}
\EA$
ใๅพใพใใ
$(0,0,0,0)$ใใ$(2,2,-1,-1)$ใใค้ฒใใใจใง
$\BA\D\\
\sum_{n=1}^\infty \frac{1}{n^2-x^2}
=\sum_{n=1}^\infty \frac{21n^3-8n^2-x^2(9n-2)}{n\binom{2n}{n}}\frac{(1-x,1+x,1-2x,1+2x)_{n-1}}{(1-x,1+x)_{2n}}
\EA$
ใๅพใพใใ
$\rm Koecher's~identity$
$\quad$ๆฌกใฎ็ญๅผใๆใ็ซใกใพใใ
$\BA\D\\
\sum_{n=1}^\infty \frac{1}{n(n^2-x^2)}=\frac{1}{2}\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^3\binom{2n}{n}}\L(1+\frac{4n^2}{n^2-x^2}\R)\prod_{m=1}^{n-1}\L(1-\frac{x^2}{m^2}\R)
\EA$
ใใใฏ๏ผ
$\BA\D\\
F(n,k)&=\frac{(-1)^nk!(1-x,1+x)_n}{(2n+k+1)!((n+k+1)^2-x^2)}\\
G(n,k)&=\frac{(-1)^nk!(1-x,1+x)_n(5(n+1)^2-x^2+k^2+4k(n+1))}{(2n+k+2)!((n+k+1)^2-x^2)(2n+2)}
\EA$
ใ
$\BA\D\\
F(n,k)-F(n+1,k)=G(n,k)-G(n,k+1)
\EA$
ใๆบใใใใจใใ
$\BA\D\\
\sum_{n=0}^\infty F(0,n)=\sum_{n=0}^\infty G(n,0)
\EA$
ใ่ใใใใจใงๅพใใใจใใงใใพใใใใใๅซใใฆไผผใ็ญๅผใฎ่จผๆใ๏ผไพใใฐ
ใใ
ใง่ฆใใใจใใงใใพใใ
$\quad$ใงใฏใใฎๅผใ${\rm WZ~}{n\textrm{-tuple}}$ใจใใฆ้ๆฅ็ใซ่จผๆใใฆใฟใพใใ
$\BA\D\\
H(i,j,k,l)=\frac{(a_1+a_3)_{i+k}(a_1+a_4)_{i+l}(a_2+a_3)_{j+k}(a_2+a_4)_{j+l}}{(a_1)_i(a_2)_j(a_3)_k(a_4)_l(-1+a_1+a_2+a_3+a_4)_{i+j+k+l+1}}
\EA$
ใซใใใฆ๏ผ$k=0$ใฎๅ ดๅใฎ
$\BA\D\\
H(i,j,0,l)=\frac{(a_1+a_3)_{i}(a_1+a_4)_{i+l}(a_2+a_3)_{j}(a_2+a_4)_{j+l}}{(a_1)_i(a_2)_j(a_4)_l(-1+a_1+a_2+a_3+a_4)_{i+j+l+1}}
\EA$
ใซๅฏพใใฆ๏ผ$(a_1,a_2,a_3,a_4)=(1-x,1,0,x-y)$ใไปฃๅ
ฅใ๏ผ$l\to k$ใจๆธใ
$\BA\D\\
&H(i,j,k)=\frac{(1-y)_{i+k}(1+x-y)_{j+k}}{(x-y)_k(1-y)_{i+j+k+1}}\\
&F_1(i,j,k)=\frac{H(i,j,k)}{1-x+i}\\
&F_2(i,j,k)=\frac{H(i,j,k)}{j+1}\\
&F_3(i,j,k)=-\frac{H(i,j,k)}{x-y+k}
\EA$
ใจๆนใใพใใ
$\quad (0,0,0)$ใๅง็นใจใใฆ$(1,0,0)$ใใค้ฒใใ ใจใใฎ้ใฟใฎๅใฏ
$\BA\D\\
\sum_{n=0}^\infty F_1(n,0,0)=\sum_{n=1}^\infty \frac{1}{(n-x)(n-y)}
\EA$
ใจใชใใพใใ
$\quad(0,0,0)$ใๅง็นใจใใฆ$(1,2,-1)$ใใค้ฒใใ ใจใใฎ้ใฟใฎๅใฏ
$\BA\D\\
&\sum_{n=0}^\infty (F_1(n,2n,-n)+F_2(n+1,2n,-n)+F_2(n+1,2n+1,-n)-F_3(n+1,2n+2,-n-1))\\
=&\sum_{n=0}^\infty
\L(\frac{1}{1-x+n}\frac{(1+x-y)_{n}}{(x-y)_{-n}(1-y)_{2n+1}}
+\frac{1}{2n+1}\frac{(1-y)(1+x-y)_{n}}{(x-y)_{-n}(1-y)_{2n+2}}
+\frac{1}{2n+2}\frac{(1-y)(1+x-y)_{n+1}}{(x-y)_{-n}(1-y)_{2n+3}}
+\frac{1}{x-y-n-1}\frac{(1+x-y)_{n+1}}{(x-y)_{-n-1}(1-y)_{2n+3}}
\R)\\
=&\sum_{n=1}^\infty \L(\frac{2n-y}{n-x}+\frac{1-y}{2n-1}+\frac{n+x-y}{2n}\R)\frac{(-1)^{n-1}(1-x+y,1+x-y)_{n-1}}{(1-y)_{2n}}
\EA$
ใจใชใใพใใใใชใใก
$\BA\D\\
\sum_{n=1}^\infty \frac{1}{(n-x)(n-y)}
=\sum_{n=1}^\infty \L(\frac{2n-y}{n-x}+\frac{1-y}{2n-1}+\frac{n+x-y}{2n}\R)\frac{(-1)^{n-1}(1-x+y,1+x-y)_{n-1}}{(1-y)_{2n}}
\EA$
ใๆใ็ซใกใพใใ$y\to 0$ใจใใใจ
$\BA\D\\
{[\rm A]}\qquad\qquad
\sum_{n=1}^\infty \frac{1}{n(n-x)}
=\sum_{n=1}^\infty \L(\frac{2n}{n-x}+\frac{1}{2n-1}+\frac{n+x}{2n}\R)\frac{(-1)^{n-1}(1-x,1+x)_{n-1}}{(2n)!}
\EA$
ใจใชใ๏ผ$x\to -x$ใจใใใจ
$\BA\D\\
{[\rm B]}\qquad\qquad
\sum_{n=1}^\infty \frac{1}{n(n+x)}
=\sum_{n=1}^\infty \L(\frac{2n}{n+x}+\frac{1}{2n-1}+\frac{n-x}{2n}\R)\frac{(-1)^{n-1}(1-x,1+x)_{n-1}}{(2n)!}
\EA$
ใจใชใใพใใ${[\rm A]+[\rm B]}$ใใ
$\BA\D\\
\sum_{n=1}^\infty \frac{2}{n^2-x^2}
=\sum_{n=1}^\infty \L(\frac{4n^2}{n^2-x^2}+\frac{2n+1}{2n-1}\R)\frac{(-1)^{n-1}}{n^2\binom{2n}{n}}\prod_{m=1}^{n-1}\L(1-\frac{x^2}{m^2}\R)
\EA$
ใพใ๏ผ${[\rm A]-[\rm B]}$ใใ
$\BA\D\\
\sum_{n=1}^\infty \frac{2}{n(n^2-x^2)}
=\sum_{n=1}^\infty \L(\frac{4n}{n^2-x^2}+\frac{1}{n}\R)\frac{(-1)^{n-1}}{n^2\binom{2n}{n}}\prod_{m=1}^{n-1}\L(1-\frac{x^2}{m^2}\R)
\EA$
ใๅพใพใใใใใฏ$\rm Koecher's~identity$ใซ็ญใใใงใใ
$\BA\D\\
\EA$
$\D
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