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4 sources with state-of-art information about this topic on some recent Polynomial identities for $\pi^c$ are:

1.- "Rational Hypergeometric Ramanujan Identities for $1/π^c$: Survey and Generalizations" (2021) arXiv:2101.12592 by J. Guillera and H. Cohen.

2.- There is also a connection of Hypergeometric Motives for some $1/\pi^2$ identities to Hilbert Modular Forms in this work "Special Hypergeometric Motives and Their L-Functions: Asai Recognition" by : Lassina Dembélé, Alexei Panchishkin, John Voight & Wadim Zudilin (2020). Experimental Mathematics, DOI: 10.1080/10586458.2020.1737990

3.- K.C. Au takes the Wilf Zeilberger method to another level for proving some long time conjectured $\pi^c$ hypergeometric-type identities with $c=-3,-4,4$. Kam Cheong Au. "Wilf-Zeiberger Seeds and Non-Trivial Hypergeometric Identities". (Dec.2023) arXiv:2312.14051v2

Proofs of Cullen's, Guillera's, Zhao's, Gourevitch, Zhi-Wei Sun identities, among others, are found in this work.

4.- Jesus Guillera once said that he thought all Ramanujan-type Pi formulas, could be eventually proven by means of the WZ algorithm. Inspired on this, and on 2.- and 3.- as well, I have recently placed a question here in MO since I think there should be a link between Hypergeometric Motives (the connection to L-functions and Modular forms) and the WZ algorithm. I think this topic has not already been deeply studied .

At the end of this question I have placed some recently found and proven (Apr. 2024) high weight $\pi^c$ ($c=-4,4$) and $\zeta(5)$ new hypergeometric-type identities.

4 sources with state-of-art information about this topic on some recent Polynomial identities for $\pi^c$ are:

1.- "Rational Hypergeometric Ramanujan Identities for $1/π^c$: Survey and Generalizations" (2021) arXiv:2101.12592 by J. Guillera and H. Cohen.

2.- There is also a connection of Hypergeometric Motives for some $1/\pi^2$ identities to Hilbert Modular Forms in this work "Special Hypergeometric Motives and Their L-Functions: Asai Recognition" by : Lassina Dembélé, Alexei Panchishkin, John Voight & Wadim Zudilin (2020). Experimental Mathematics, DOI: 10.1080/10586458.2020.1737990

3.- K.C. Au takes the Wilf Zeilberger method to another level for proving some long time conjectured $\pi^c$ hypergeometric-type identities with $c=-3,-4,4$. Kam Cheong Au. "Wilf-Zeiberger Seeds and Non-Trivial Hypergeometric Identities". (Dec.2023) arXiv:2312.14051v2

Proofs of Cullen's, Guillera's, Zhao's, Gourevitch, Zhi-Wei Sun identities, among others, are found in this work.

4.- Jesus Guillera once said that he thought all Ramanujan-type Pi formulas, could be eventually proven by means of the WZ algorithm. Inspired on this, and on 2.- and 3.- as well, I have recently placed a question here in MO since I think there should be a link between Hypergeometric Motives (the connection to L-functions and Modular forms) and the WZ algorithm. I think this topic has not already been deeply studied .

At the end of this question, in the EXAMPLES section, I have placed some recently found and proven (Apr. 2024) $\pi^{±4}$ and $\zeta(5)$ new hypergeometric-type identities.

4 sources with state-of-art information about this topic on some recent Polynomial identities for $\pi^c$ are:

1.- "Rational Hypergeometric Ramanujan Identities for $1/π^c$: Survey and Generalizations" (2021) arXiv:2101.12592 by J. Guillera and H. Cohen.

2.- There is also a connection of Hypergeometric Motives for some $1/\pi^2$ identities to Hilbert Modular Forms in this work "Special Hypergeometric Motives and Their L-Functions: Asai Recognition" by : Lassina Dembélé, Alexei Panchishkin, John Voight & Wadim Zudilin (2020). Experimental Mathematics, DOI: 10.1080/10586458.2020.1737990

3.- K.C. Au takes the Wilf Zeilberger method to another level for proving some long time conjectured $\pi^c$ hypergeometric-type identities with $c=-3,-4,4$. Kam Cheong Au. "Wilf-Zeiberger Seeds and Non-Trivial Hypergeometric Identities". (Dec.2023) arXiv:2312.14051v2

Proofs of Cullen's, Guillera's, Zhao's, Gourevitch, Zhi-Wei Sun identities, among others, are found in this work.

4.- Jesus Guillera once said that he thought all Ramanujan-type Pi formulas, could be eventually proven by means of the WZ algorithm. Inspired on this, and on 2.- and 3.- as well, I have recently placed a question here in MO since I think there should be a link between Hypergeometric Motives (the connection to L-functions and Modular forms) and the WZ algorithm. I think this topic has not already been deeply studied .

At the end of this question I have placed some recently found and proven (Apr. 2024) high weight $\pi^c$ ($c=-4,4$) and $\zeta(5)$ hypergeometric-type identities.

4 sources with state-of-art information about this topic on some recent Polynomial identities for $\pi^c$ are:

1.- "Rational Hypergeometric Ramanujan Identities for $1/π^c$: Survey and Generalizations" (2021) arXiv:2101.12592 by J. Guillera and H. Cohen.

2.- There is also a connection of Hypergeometric Motives for some $1/\pi^2$ identities to Hilbert Modular Forms in this work "Special Hypergeometric Motives and Their L-Functions: Asai Recognition" by : Lassina Dembélé, Alexei Panchishkin, John Voight & Wadim Zudilin (2020). Experimental Mathematics, DOI: 10.1080/10586458.2020.1737990

3.- K.C. Au takes the Wilf Zeilberger method to another level for proving some long time conjectured $\pi^c$ hypergeometric-type identities with $c=-3,-4,4$. Kam Cheong Au. "Wilf-Zeiberger Seeds and Non-Trivial Hypergeometric Identities". (Dec.2023) arXiv:2312.14051v2

Proofs of Cullen's, Guillera's, Zhao's, Gourevitch, Zhi-Wei Sun identities, among others, are found in this work.

4.- Jesus Guillera once said that he thought all Ramanujan-type Pi formulas, could be eventually proven by means of the WZ algorithm. Inspired on this, and on 2.- and 3.- as well, I have recently placed a question here in MO since I think there should be a link between Hypergeometric Motives (the connection to L-functions and Modular forms) and the WZ algorithm. I think this topic has not already been deeply studied .

At the end of this question I have placed some recently found and proven (Apr. 2024) high weight $\pi^c$ ($c=-4,4$) and $\zeta(5)$ new hypergeometric-type identities.

4 sources with state-of-art information about this topic on some recent Polynomial identities for $\pi^c$ are:

1.- "Rational Hypergeometric Ramanujan Identities for $1/π^c$: Survey and Generalizations" (2021) arXiv:2101.12592 by J. Guillera and H. Cohen.

2.- There is also a connection of Hypergeometric Motives for some $1/\pi^2$ identities to Hilbert Modular Forms in this work "Special Hypergeometric Motives and Their L-Functions: Asai Recognition" by : Lassina Dembélé, Alexei Panchishkin, John Voight & Wadim Zudilin (2020). Experimental Mathematics, DOI: 10.1080/10586458.2020.1737990

3.- K.C. Au takes the Wilf Zeilberger method to another level for proving some long time conjectured $\pi^c$ hypergeometric-type identities with $c=-3,-4,4$. Kam Cheong Au. "Wilf-Zeiberger Seeds and Non-Trivial Hypergeometric Identities". (Dec.2023) arXiv:2312.14051v2

Proofs of Cullen's, Guillera's, Zhao's and Gourevitch's identities are found in this work.

4.- Jesus Guillera once said that he thought all Ramanujan-type Pi formulas, could be eventually proven by means of the WZ algorithm. Inspired on this, and on 2.- and 3.- as well, I have recently placed a question here in MO since I think there should be a link between Hypergeometric Motives (the connection to L-functions and Modular forms) and the WZ algorithm. I think this topic has not already been deeply studied .

At the end of this question I have placed some recently found and proven (Apr. 2024) high weight $\pi^c$ ($c=-4,4$) and $\zeta(5)$ hypergeometric-type identities.

4 sources with state-of-art information about this topic on some recent Polynomial identities for $\pi^c$ are:

1.- "Rational Hypergeometric Ramanujan Identities for $1/π^c$: Survey and Generalizations" (2021) arXiv:2101.12592 by J. Guillera and H. Cohen.

2.- There is also a connection of Hypergeometric Motives for some $1/\pi^2$ identities to Hilbert Modular Forms in this work "Special Hypergeometric Motives and Their L-Functions: Asai Recognition" by : Lassina Dembélé, Alexei Panchishkin, John Voight & Wadim Zudilin (2020). Experimental Mathematics, DOI: 10.1080/10586458.2020.1737990

3.- K.C. Au takes the Wilf Zeilberger method to another level for proving some long time conjectured $\pi^c$ hypergeometric-type identities with $c=-3,-4,4$. Kam Cheong Au. "Wilf-Zeiberger Seeds and Non-Trivial Hypergeometric Identities". (Dec.2023) arXiv:2312.14051v2

Proofs of Cullen's, Guillera's, Zhao's, Gourevitch, Zhi-Wei Sun identities, among others, are found in this work.

4.- Jesus Guillera once said that he thought all Ramanujan-type Pi formulas, could be eventually proven by means of the WZ algorithm. Inspired on this, and on 2.- and 3.- as well, I have recently placed a question here in MO since I think there should be a link between Hypergeometric Motives (the connection to L-functions and Modular forms) and the WZ algorithm. I think this topic has not already been deeply studied .

At the end of this question I have placed some recently found and proven (Apr. 2024) high weight $\pi^c$ ($c=-4,4$) and $\zeta(5)$ hypergeometric-type identities.