Return to Question

Following these MO posts from Timothy Chow and Henri Cohen and their comments, based on Saha & Sinha's String Theory article, I have found three series for $\pi$, Apéry's constant $\zeta(3)$ and the Lemniscate constant $2\varpi$. Their summands have non-linear (rational or algebraic) Gamma Function or Pochhammer's arguments which makes them very atypical.

All series are slowly convergent but can be accelerated applying $\Delta$-Euler-MacLaurin Method by Henri Cohen which is implemented in PARI-GP script sumnumdelta() as it is seen below.

Will it be possible to prove them?

I.- $\pi$ series $$\pi=\small{2\sinh\small{\left(\frac{\pi}{2}\right)}\cdot\sum_{n=1}^\infty\frac{1}{n!}\cdot\small{\left(1+\frac{1}{4n}\right)_{n-1}}\frac{2n+1}{(2n-1)(4n^2+1)}}$$

II.- Apéry's constant series with a free parameter. For $\phi_n(t)=-\frac{n\,t}{n+t}$ with $t\in\mathbb{C}$ and $n\in\mathbb{N}$ $$\small{\zeta(3)}=\small{\sum_{n=1}^\infty\frac{\small{\left(\phi_n(t)+1\right)_n}}{n^3\,n!}\cdot\small{\left(\frac{n+t}{n}+t\cdot\frac{(n+2t)\,\left[\Psi(\phi_n(t)+n)-\Psi(\phi_n(t)+1)\right]}{n+t}\right)}}$$ where $\small{\Psi(x)}$ is the Digamma function and the numerator difference represents a harmonic function with rational arguments. Note that $t=0$ gives the classical $\zeta(3)$ series.

III.- The Lemniscate constant, a trascendental number, see Section 6.1 in

Finch, Steven R., Mathematical constants, Encyclopedia of Mathematics and Its Applications 94. Cambridge: Cambridge University Press (ISBN 0-521-81805-2/hbk). xx, 602 p. (2003). ZBL1054.00001.$$\small{2\varpi} =\small{\frac{\;\Gamma(\frac{1}{4})^2}{\sqrt{2\pi}}=5.2441151085842396209296791797822388273655...}$$ has the following strange series with algebraic Pochhammer's arguments for its squared value $$\small{(2\varpi)^2}=\small{\pi\cdot\left(8+\sum_{n=1}^\infty\small{\left(\frac{1}{4n-2}-\frac{1}{4n}+\frac{2}{4n+1}\right)}\cdot\frac{\left(1-\frac{n}{2}+\frac{n}{2}\sqrt{1-\frac{1}{8n^3}}\right)_{n-1}^2}{n!^2}\right)}$$

Q: It is unknown if there is any proof of these identities. Can a variation of Rosengren's proof be applied for these cases?

Following these MO posts from Timothy Chow and Henri Cohen and their comments, based on Saha & Sinha's String Theory article, I have found three series for $\pi$, Apéry's constant $\zeta(3)$ and the Lemniscate constant $2\varpi$. Their summands have non-linear (rational or algebraic) Gamma Function or Pochhammer's arguments which makes them very atypical.

All series are slowly convergent but can be accelerated applying $\Delta$-Euler-MacLaurin Method by Henri Cohen which is implemented in PARI-GP script sumnumdelta() as it is seen below.

Will it be possible to prove them?

I.- $\pi$ series $$\pi=\small{2\sinh\small{\left(\frac{\pi}{2}\right)}\cdot\sum_{n=1}^\infty\frac{1}{n!}\cdot\small{\left(1+\frac{1}{4n}\right)_{n-1}}\frac{2n+1}{(2n-1)(4n^2+1)}}$$

II.- Apéry's constant series with a free parameter. For $\phi_n(t)=-\frac{n\,t}{n+t}$ with $t\in\mathbb{C}$ and $n\in\mathbb{N}$ $$\small{\zeta(3)}=\small{\sum_{n=1}^\infty\frac{\small{\left(\phi_n(t)+1\right)_n}}{n^3\,n!}\cdot\small{\left(\frac{n+t}{n}+t\cdot\frac{(n+2t)\,\left[\psi\left(\phi_n(t)+n\right)-\psi\left(\phi_n(t)+1\right)\right]}{n+t}\right)}}$$ where $\small{\psi(x)}$ is the Digamma function and the numerator difference represents a harmonic function with rational arguments. Note that $t=0$ gives the classical $\zeta(3)$ series.

III.- The Lemniscate constant, a trascendental number, see Section 6.1 in

Finch, Steven R., Mathematical constants, Encyclopedia of Mathematics and Its Applications 94. Cambridge: Cambridge University Press (ISBN 0-521-81805-2/hbk). xx, 602 p. (2003). ZBL1054.00001.$$\small{2\varpi} =\small{\frac{\;\Gamma(\frac{1}{4})^2}{\sqrt{2\pi}}=5.2441151085842396209296791797822388273655...}$$ has the following strange series with algebraic Pochhammer's arguments for its squared value $$\small{(2\varpi)^2}=\small{\pi\cdot\left(8+\sum_{n=1}^\infty\small{\left(\frac{1}{4n-2}-\frac{1}{4n}+\frac{2}{4n+1}\right)}\cdot\frac{\left(1-\frac{n}{2}+\frac{n}{2}\sqrt{1-\frac{1}{8n^3}}\right)_{n-1}^2}{n!^2}\right)}$$

Q: It is unknown if there is any proof of these identities. Can a variation of Rosengren's proof be applied for these cases?

Following these MO posts from Timothy Chow and Henri Cohen and their comments, based on Saha & Sinha's String Theory article, I have found three series for $\pi$, Apéry's constant $\zeta(3)$ and the Lemniscate constant $2\varpi$. Their summands have non-linear (rational or algebraic) Gamma Function or Pochhammer's arguments which makes them very atypical.

Will it be possible to prove them?

I.- $\pi$ series $$\pi=\small{2\sinh\small{\left(\frac{\pi}{2}\right)}\cdot\sum_{n=1}^\infty\frac{1}{n!}\cdot\small{\left(1+\frac{1}{4n}\right)_{n-1}}\frac{2n+1}{(2n-1)(4n^2+1)}}$$

II.- Apéry's constant series with a free parameter. For $\phi_n(t)=-\frac{n\,t}{n+t}$ with $t\in\mathbb{C}$ and $n\in\mathbb{N}$ $$\small{\zeta(3)}=\small{\sum_{n=1}^\infty\frac{\small{\left(\phi_n(t)+1\right)_n}}{n^3\,n!}\cdot\small{\left(\frac{n+t}{n}+t\cdot\frac{(n+2t)\,\left[\Psi(\phi_n(t)+n)-\Psi(\phi_n(t)+1)\right]}{n+t}\right)}}$$ where $\small{\Psi(x)}$ is the Digamma function and the numerator difference represents a harmonic function with rational arguments. Note that $t=0$ gives the classical $\zeta(3)$ series.

III.- The Lemniscate constant, a trascendental number, see Section 6.1 in

Finch, Steven R., Mathematical constants, Encyclopedia of Mathematics and Its Applications 94. Cambridge: Cambridge University Press (ISBN 0-521-81805-2/hbk). xx, 602 p. (2003). ZBL1054.00001.$$\small{2\varpi} =\small{\frac{\;\Gamma(\frac{1}{4})^2}{\sqrt{2\pi}}=5.2441151085842396209296791797822388273655...}$$ has the following strange series with algebraic Pochhammer's arguments for its squared value $$\small{(2\varpi)^2}=\small{\pi\cdot\left(8+\sum_{n=1}^\infty\small{\left(\frac{1}{4n-2}-\frac{1}{4n}+\frac{2}{4n+1}\right)}\cdot\frac{\left(1-\frac{n}{2}+\frac{n}{2}\sqrt{1-\frac{1}{8n^3}}\right)_{n-1}^2}{n!^2}\right)}$$

Q: It is unknown if there is any proof of these identities. Can a variation of Rosengren's proof be applied for these cases?

Following these MO posts from Timothy Chow and Henri Cohen and their comments, based on Saha & Sinha's String Theory article, I have found three series for $\pi$, Apéry's constant $\zeta(3)$ and the Lemniscate constant $2\varpi$. Their summands have non-linear (rational or algebraic) Gamma Function or Pochhammer's arguments which makes them very atypical.

All series are slowly convergent but can be accelerated applying $\Delta$-Euler-MacLaurin Method by Henri Cohen which is implemented in PARI-GP script sumnumdelta() as it is seen below.

Will it be possible to prove them?

I.- $\pi$ series $$\pi=\small{2\sinh\small{\left(\frac{\pi}{2}\right)}\cdot\sum_{n=1}^\infty\frac{1}{n!}\cdot\small{\left(1+\frac{1}{4n}\right)_{n-1}}\frac{2n+1}{(2n-1)(4n^2+1)}}$$

II.- Apéry's constant series with a free parameter. For $\phi_n(t)=-\frac{n\,t}{n+t}$ with $t\in\mathbb{C}$ and $n\in\mathbb{N}$ $$\small{\zeta(3)}=\small{\sum_{n=1}^\infty\frac{\small{\left(\phi_n(t)+1\right)_n}}{n^3\,n!}\cdot\small{\left(\frac{n+t}{n}+t\cdot\frac{(n+2t)\,\left[\Psi(\phi_n(t)+n)-\Psi(\phi_n(t)+1)\right]}{n+t}\right)}}$$ where $\small{\Psi(x)}$ is the Digamma function and the numerator difference represents a harmonic function with rational arguments. Note that $t=0$ gives the classical $\zeta(3)$ series.

III.- The Lemniscate constant, a trascendental number, see Section 6.1 in

Finch, Steven R., Mathematical constants, Encyclopedia of Mathematics and Its Applications 94. Cambridge: Cambridge University Press (ISBN 0-521-81805-2/hbk). xx, 602 p. (2003). ZBL1054.00001.$$\small{2\varpi} =\small{\frac{\;\Gamma(\frac{1}{4})^2}{\sqrt{2\pi}}=5.2441151085842396209296791797822388273655...}$$ has the following strange series with algebraic Pochhammer's arguments for its squared value $$\small{(2\varpi)^2}=\small{\pi\cdot\left(8+\sum_{n=1}^\infty\small{\left(\frac{1}{4n-2}-\frac{1}{4n}+\frac{2}{4n+1}\right)}\cdot\frac{\left(1-\frac{n}{2}+\frac{n}{2}\sqrt{1-\frac{1}{8n^3}}\right)_{n-1}^2}{n!^2}\right)}$$

Q: It is unknown if there is any proof of these identities. Can a variation of Rosengren's proof be applied for these cases?

Following these MO posts from Timothy Chow and Henri Cohen and their comments, based on Saha & Sinha's String Theory article, I have found three strange series for $\pi$, Apéry's constant $\zeta(3)$ and the Lemniscate constant $2\varpi$. Their summands have non-linear (rational or algebraic) Gamma Function or Pochhammer's arguments which makes them very atypical.

Will it be possible to prove them?

I.- $\pi$ series $$\pi=\small{2\sinh\small{\left(\frac{\pi}{2}\right)}\cdot\sum_{n=1}^\infty\frac{1}{n!}\cdot\small{\left(1+\frac{1}{4n}\right)_{n-1}}\frac{2n+1}{(2n-1)(4n^2+1)}}$$

II.- Apéry's constant series with a free parameter. For $\phi_n(t)=-\frac{n\,t}{n+t}$ with $t\in\mathbb{C}$ and $n\in\mathbb{N}$ $$\small{\zeta(3)}=\small{\sum_{n=1}^\infty\frac{\small{\left(\phi_n(t)+1\right)_n}}{n^3\,n!}\cdot\small{\left(\frac{n+t}{n}+t\cdot\frac{(n+2t)\,\left[\Psi(\phi_n(t)+n)-\Psi(\phi_n(t)+1)\right]}{n+t}\right)}}$$ where $\small{\Psi(x)}$ is the Digamma function and the numerator difference represents a harmonic function with rational arguments. Note that $t=0$ gives the classical $\zeta(3)$ series.

III.- The Lemniscate constant, a trascendental number, see Section 6.1 in

Finch, Steven R., Mathematical constants, Encyclopedia of Mathematics and Its Applications 94. Cambridge: Cambridge University Press (ISBN 0-521-81805-2/hbk). xx, 602 p. (2003). ZBL1054.00001.$$\small{2\varpi} =\small{\frac{\;\Gamma(\frac{1}{4})^2}{\sqrt{2\pi}}=5.2441151085842396209296791797822388273655...}$$ has the following strange series with algebraic Pochhammer's arguments for its squared value $$\small{(2\varpi)^2}=\small{\pi\cdot\left(8+\sum_{n=1}^\infty\small{\left(\frac{1}{4n-2}-\frac{1}{4n}+\frac{2}{4n+1}\right)}\cdot\frac{\left(1-\frac{n}{2}+\frac{n}{2}\sqrt{1-\frac{1}{8n^3}}\right)_{n-1}^2}{n!^2}\right)}$$

Q: It is unknown if there is any proof of these identities. Can a variation of Rosengren's proof be applied for these cases?

Following these MO posts from Timothy Chow and Henri Cohen and their comments, based on Saha & Sinha's String Theory article, I have found three series for $\pi$, Apéry's constant $\zeta(3)$ and the Lemniscate constant $2\varpi$. Their summands have non-linear (rational or algebraic) Gamma Function or Pochhammer's arguments which makes them very atypical.

Will it be possible to prove them?

I.- $\pi$ series $$\pi=\small{2\sinh\small{\left(\frac{\pi}{2}\right)}\cdot\sum_{n=1}^\infty\frac{1}{n!}\cdot\small{\left(1+\frac{1}{4n}\right)_{n-1}}\frac{2n+1}{(2n-1)(4n^2+1)}}$$

II.- Apéry's constant series with a free parameter. For $\phi_n(t)=-\frac{n\,t}{n+t}$ with $t\in\mathbb{C}$ and $n\in\mathbb{N}$ $$\small{\zeta(3)}=\small{\sum_{n=1}^\infty\frac{\small{\left(\phi_n(t)+1\right)_n}}{n^3\,n!}\cdot\small{\left(\frac{n+t}{n}+t\cdot\frac{(n+2t)\,\left[\Psi(\phi_n(t)+n)-\Psi(\phi_n(t)+1)\right]}{n+t}\right)}}$$ where $\small{\Psi(x)}$ is the Digamma function and the numerator difference represents a harmonic function with rational arguments. Note that $t=0$ gives the classical $\zeta(3)$ series.

III.- The Lemniscate constant, a trascendental number, see Section 6.1 in

Finch, Steven R., Mathematical constants, Encyclopedia of Mathematics and Its Applications 94. Cambridge: Cambridge University Press (ISBN 0-521-81805-2/hbk). xx, 602 p. (2003). ZBL1054.00001.$$\small{2\varpi} =\small{\frac{\;\Gamma(\frac{1}{4})^2}{\sqrt{2\pi}}=5.2441151085842396209296791797822388273655...}$$ has the following strange series with algebraic Pochhammer's arguments for its squared value $$\small{(2\varpi)^2}=\small{\pi\cdot\left(8+\sum_{n=1}^\infty\small{\left(\frac{1}{4n-2}-\frac{1}{4n}+\frac{2}{4n+1}\right)}\cdot\frac{\left(1-\frac{n}{2}+\frac{n}{2}\sqrt{1-\frac{1}{8n^3}}\right)_{n-1}^2}{n!^2}\right)}$$

Q: It is unknown if there is any proof of these identities. Can a variation of Rosengren's proof be applied for these cases?