If $\small{(F,G)}$ is a WZ-pair and general asymptotic conditions $\lim_{k\rightarrow\infty}\small{G(n,k)=0}$ and $\lim_{n\rightarrow\infty}\small{F(n,k)=0}$ hold, then we have the certified hypergeometric identity $$\sum_{k=0}^\infty \small{F(0,k)}=\sum_{n=0}^\infty \small{G(n,0)}$$
Isolating the hypergeometric factor of each summand we get two hypergeometric motives $\small{HGM_F}$ and $\small{HGM_G}$. The question follows straightforward. How these data are related?. Do they share a connection?. Does the WZ algorithm generates a link between the respective Hodge vectors?.
Well, this is the simpler part of the question. We can go some steps further.
If $\small{(F,G)}$ is a WZ-pair and the above identity holds, we search another WZ-Pair $\small{(F_{s,t},G_{s,t})}$ such that for some $s\in\mathbb{N},\,\,t\in\mathbb{Z}$, $$\small{F_{s,t}(n,k)=F(sn,k+tn)}$$. If, in addition $t\in\mathbb{Z_{\ge 0}}$, we have these identities (null sums are 0) $$\small{G_{s,t}(n,k)=\sum_{m=0}^{s-1}G(sn+m,k+tn)+\sum_{j=0}^{t-1}F(sn+s,j+k+tn)}$$ $$\small{G_{s,t}(n,k)=\sum_{m=0}^{s-1}G(sn+m,k+t+tn)+\sum_{j=0}^{t-1}F(sn,j+k+tn)}$$ If the new asymptotic conditions (limit zero conditions) hold and this WZ Pair exists then it is certified that $$\small{\sum_{n=0}^\infty} \small{G(n,0)}=\small{\sum_{k=0}^\infty}\small{F(0,k)}=\sum_{k=0}^\infty \small{F_{s,t}(0,k)}=\sum_{n=0}^\infty \small{G_{s,t}(n,0)}$$ and we formulate the same questions but now between $\small{HGM_G}$ and $\small{HGM_{G_{s,t}}}$, the hypergeometric motive data associated to $\small{G(n,0)}$ and $\small{G_{s,t}(n,0)}$ respectively.
I am not aware if this topic on how the WZ algorithm transforms hypergeometric motives has been studied. Any reference will be appreciated.
We name this $(s,t)$ transformation from $\small{HGM_G}$ to $\small{HGM_{G_{s,t}}}$as WZ$_{s,t}$, a process that we applied to find some new non-trivial hypergeometric identities (whose hypergeometric motives over $\mathbb{Q}$ are far beyond the currently known LMFDB data base). We write it as WZ$_{s,t}\,\small{:HGM_G\longrightarrow HGM_{G_{s,t}}}$
We use this notation for hypergeometric identities of constant $\omega$ $$\small{\omega=c\cdot\small{\sum_{n=0}^\infty}\small{\frac{P(n)}{Q(n)}}}\cdot\small{HGM(n)}$$ where $\frac{P(n)}{Q(n)}$ is the rational part (a ratio of polynomials) and $\small{HGM(n)}$ the hypergeometric motive factor $$\small{HGM(n)=\rho^n\cdot\frac{(\alpha_1)_n(\alpha_2)_n...(\alpha_p)_n}{(\beta_1)_n(\beta_2)_n...(\beta_p)_n} }$$ associated to $\small{HGM=\textbf{[}\alpha,\beta\,|\,1/\rho\textbf{]}}$ data, with $\small{\rho\in{ℚ}}$ and $\small{\alpha=\left\{\alpha_1,\alpha_2...,\alpha_p\right\}}$, $\small{\beta=\left\{\beta_1,\beta_2...,\beta_p\right\}\in{ℚ^p}}$ are two equal size disjoint multisets where all irreducible fractions $\ell/q \in (0,1]$ with denominator $q$ occur and $\gcd(\ell,q) = 1$ being $0 < \ell < q$ or $\ell=q=1$. Sum factor $c$ could be algebraic, but we have only considered constants $\omega$ identities with $c\in\mathbb{Q}$.
For a more succinct notation we also use cyclotomic parameters to define hypergeometric motive data over $ℚ$ as $\small{HGM=\textbf{[}CP\,|\,z_0\textbf{]}}$ where $\small{CP=q_1^{\nu_1}q_2^{\nu_2}...q_m^{\nu_m}},\,\,\small{q_i\in{ℕ},\,\nu_i\in{ℤ/\{0\}},\,i=1,2,..,m}$ and the specialization point $\small{z_0=1/\rho\neq\{0,1\}}$.
EXAMPLES
These are three examples where this WZ$_{s,t}$ transformation is applied. Non-trivial hypergeometric identities are taken from K.C. Au 2023 work to derive other new and proven identities. (There is no room here to place the corresponding WZ certificates).
I.- Example IV, Guillera's formula for $\pi^4,\, \small{HGM_G=[1^72^{-5}3^{-1}\,|\,27/4]}$
I.1.- WZ$_{1,-1}:\small{[1^72^{-5}3^{-1}\,|\,27/4]}\longrightarrow\small{[1^32^13^14^{-3}\,|\,-256/27]}$
$$\small{\pi^4}=\small{32\cdot\sum_{n=0}^\infty\frac{P(n)}{(n+1)(2n+1)^2(4n+1)^4(4n+3)^4}\cdot\left(-\frac{27}{256}\right)^n\cdot\frac{(1)_n^3\,(\frac{1}{2})_n\,(\frac{1}{3})_n\,(\frac{2}{3})_n}{\,(\frac{1}{4})_n^3\,(\frac{3}{4})_n^3}}$$ $$\small{P(n)=9056\,n^7 + 40856\,n^6 + 77616\,n^5 + 80176\,n^4 + 48412\,n^3 + 16998\,n^2 + 3197\,n + 248} $$
I.2.- WZ$_{1,1}:\small{[1^72^{-5}3^{-1}\,|\,27/4]}\longrightarrow\small{[1^72^55^13^{-1}4^{-3}6^{-4}\,|\,-3^{15}/5^5]}$
$$\small{\pi^4}=\small{\frac{32}{243}\cdot\sum_{n=0}^\infty\frac{P(n)}{Q(n)}\cdot\left(-\frac{5}{27}\right)^{5n}\cdot\frac{(1)_n^7\,(\frac{1}{2})_n^5\,(\frac{1}{5})_n\,(\frac{2}{5})_n\,(\frac{3}{5})_n\,(\frac{4}{5})_n}{(\frac{1}{3})_n\,(\frac{2}{3})_n\,(\frac{1}{4})_n^3\,(\frac{3}{4})_n^3\,(\frac{1}{6})_n^4\,(\frac{5}{6})_n^4}}$$ $$\scriptsize{P(n)=29392961536\,n^{12} + 217229156352\,n^{11} + 725653655552\,n^{10} + 1446938726912\,n^9 + 1915347808512\,n^8 \\ + 1770423856672\,n^7 + 1169764497776\,n^6 + 555679147152\,n^5 + 188016602440\,n^4 + 44113300042\,n^3 \\ + 6801570483\,n^2 + 617848010\,n + 24969216} $$ $$\scriptsize{Q(n)=(3n+1)(3n+2)(4n+1)^3(4n+3)^3(6n+1)^4(6n+5)^4}$$
II.- Example II, Zhao's formula for $1/\pi^4,\, \small{HGM_G=[1^{-9}2^53^14^1\,|\,-256/27]}$
II.1.- WZ$_{1,1}:\small{[1^{-9}2^53^14^1\,|\,-256/27]}\longrightarrow\small{[1^{-9}2^53^{-4}4^56^1\,|\,-3^9/2^{12}]}$
$$\small{\frac{1}{\pi^4}}=\small{\frac{1}{2048}\cdot\sum_{n=0}^\infty\frac{P(n)}{(3n+1)^4(3n+2)^4}\cdot\left(-\frac{16}{27}\right)^{3n}\cdot\frac{(\frac{1}{2})_n^5\,(\frac{1}{4})_n^5\,(\frac{3}{4})_n^5\,(\frac{1}{6})_n\,(\frac{5}{6})_n}{(1)_n^9\,(\frac{1}{3})_n^4\,(\frac{2}{3})_n^4}}$$ $$\scriptsize{P(n)=219147264\,n^{12} + 1003812864\,n^{11} + 2050003968\,n^{10} + 2466525696\,n^9 + 1947081600\,n^8 + 1063030336\,n^7 \\+ 412178912\,n^6 + 114629264\,n^5 + 22764136\,n^4 + 3159432\,n^3 + 291852\,n^2 + 16146\,n + 405} $$
II.2.- WZ$_{2,-1}:\small{[1^{-9}2^53^14^1\,|\,-256/27]}\longrightarrow\small{[1^{-9}2^13^{-4}4^55^16^1\,|\,-(2^43^9)/5^5]}$
$$\small{\frac{1}{\pi^4}}=\small{\frac{1}{5\cdot2^{17}}\cdot\sum_{n=0}^\infty\frac{P(n)}{Q(n)}\cdot\left(-\frac{5^5}{2^4\,3^9}\right)^{n}\cdot\frac{(\frac{1}{2})_n\,(\frac{1}{4})_n^5\,(\frac{3}{4})_n^5\,(\frac{1}{5})_n\,(\frac{2}{5})_n\,(\frac{3}{5})_n\,(\frac{4}{5})_n\,(\frac{1}{6})_n\,(\frac{5}{6})_n}{(1)_n^9\,(\frac{1}{3})_n^4\,(\frac{2}{3})_n^4}}$$ $$\scriptsize{P(n)=187595292672\,n^{12} + 827624275968\,n^{11} + 1615175479296\,n^{10} + 1839566699520\,n^9 + 1359174749952\,n^8 \\ + 685500617920\,n^7 + 241984241104\,n^6 + 60348294116\,n^5 + 10598535632\,n^4 + 1287571065\,n^3 \\ + 103611510\,n^2 + 4991895\,n + 109350} $$ $$\scriptsize{Q(n)=(3n+1)^4(3n+2)^4}$$
III.- Example X, Zhao's formula for $\zeta(5),\, \small{HGM_G=[1^{9}2^{-5}5^{-1}\,|\,-3125/1024]}$
III.1.- WZ$_{2,-1}:\small{[1^{9}2^{-5}5^{-1}\,|\,-5^5/2^{10}]}\,\longrightarrow\small{[1^{8}3^44^{-6}8^{-1}\,|\,2^{20}/3^{12}]}$
$$\small{\zeta(5)}=\small{\frac{4}{93}\cdot\sum_{n=0}^\infty\frac{P(n)}{Q(n)}\cdot\left(\frac{27}{32}\right)^{4n}\cdot\frac{(1)_n^8\,(\frac{1}{3})_n^4\,(\frac{2}{3})_n^4}{(\frac{1}{4})_n^6\,(\frac{3}{4})_n^6\,(\frac{1}{8})_n\,(\frac{3}{8})_n\,(\frac{5}{8})_n\,(\frac{7}{8})_n}}$$ $$\scriptsize{P(n)=1059092480\,n^{12} + 7972492288\,n^{11} + 27177876480\,n^{10} + 55429937664\,n^9 + 75259941120\,n^8 + 71596677120\,n^7 \\ + 48888359040\,n^6 + 24120172800\,n^5 + 8526531960\,n^4 + 2104687856\,n^3 + 344173168\,n^2 + 33469362\,n + 1463787} $$ $$\scriptsize{Q(n)=(2n+1)(4n+1)^6(4n+3)^6(8n+1)(8n+3)(8n+5)(8n+7)}$$
III.2.- WZ$_{1,1}:\small{[1^{9}2^{-5}5^{-1}\,|\,-5^5/2^{10}]}\,\longrightarrow\small{[1^{9}2^{-5}3^54^{-4}7^{-1}\,|\,-2^6\,7^7/3^{15}]}$
$$\small{\zeta(5)}=\small{\frac{8}{651}\cdot\sum_{n=0}^\infty\frac{P(n)}{Q(n)}\cdot\left(-\frac{3^{15}}{2^6\,7^7}\right)^{n}\cdot\frac{(1)_n^9\,(\frac{1}{3})_n^5\,(\frac{2}{3})_n^5}{(\frac{1}{2})_n^5\,(\frac{1}{4})_n^4\,(\frac{3}{4})_n^4\,(\frac{1}{7})_n\,(\frac{2}{7})_n\,(\frac{3}{7})_n\,(\frac{4}{7})_n\,(\frac{5}{7})_n\,(\frac{6}{7})_n}}$$ $$\scriptsize{P(n)=24969089024\,n^{14} + 204922738688\,n^{13} + 774418470912\,n^{12} + 1785479831552\,n^{11} + 2804695042304\,n^{10} \\ + 3174126528384\,n^9 + 2667904092288\,n^8 + 1691245558272\,n^7 + 812280569892\,n^6 + 294055496224\,n^5 \\ + 78965727457\,n^4 + 15251156694\,n^3 + 2002465411\,n^2 + 159994306\,n + 5869512} $$ $$\scriptsize{Q(n)=(2n+1)^5(4n+1)^4(4n+3)^4(7n+1)(7n+2)(7n+3)(7n+4)(7n+5)(7n+6)}$$
