Indirect method (associated with a certain problem of electrostatics) indicates that $$\sum\limits_{j=1}^\infty \frac{(2j-3)!!\,(2j-1)!!}{(2j-2)!!\,(2j+2)!!}=\frac{2}{3\pi}.$$ Is this result known?
Sequences and Series – Infinite Series for 1/?: Is It Known?
reference-requestsequences-and-series
Related Solutions
Using Christian Krattenthaler's hyp.m tells you the following - the important information is at the end, and the result agrees with Johannes' comment.
In[1]:= <<hyp.m
Out[1]= ▒
In[10]:= S = SUM[1/(n+1/2)*(Gamma[n]^2/Gamma[n+1/2]^2),{n,1,Infinity}]
Infinity
----- 2
\ Gamma[n]
Out[10]= > ---------------------
/ 1 1 2
----- (- + n) Gamma[- + n]
n=1 2 2
In[19]:= SF = S/.SUMF
[ 1, 1, 1 ]
| | 2
2 F | 5 3 ; 1 | Ga(1)
3 2| -, - |
[ 2 2 ]
Out[19]= --------------------------
3 2
3 Ga(-)
2
In[21]:= SF/.SListe
Be sure to apply "FOrdne" before using the following information!
2
2 S3261 Ga(1)
Out[21]= {{--------------}}
3 2
3 Ga(-)
2
In[22]:= SF/.S3261
1 1 1 3 3 1 1 1 3
[ -(-), -(-), -(-), - ] [ 1 1 ] [ -, -(-), -(-), -(-), - ]
2 | 2 2 2 2 | | -(-), -(-) | | 2 2 2 2 2 |
2 Ga(1) (Ga| | - F | 2 2 ; 1 | Ga| |)
| 3 1 1 1 | 2 1| | | 3 |
[ -(-), -, -, - ] [ 0 ] [ -(-), 1, 1, 1, 0 ]
2 2 2 2 2
Out[22]= ----------------------------------------------------------------------------------------
3 2
3 Ga(-)
2
In[23]:= SF/.S3261/.SListe
Be sure to apply "FOrdne" before using the following information!
1 1 1 3 3 1 1 1 3
[ -(-), -(-), -(-), - ] [ -, -(-), -(-), -(-), - ]
2 | 2 2 2 2 | | 2 2 2 2 2 |
2 Ga(1) (Ga| | - S2103 Ga| |)
| 3 1 1 1 | | 3 |
[ -(-), -, -, - ] [ -(-), 1, 1, 1, 0 ]
2 2 2 2 2
Out[23]= {{-------------------------------------------------------------------------}}
3 2
3 Ga(-)
2
In[33]:= R = SF/.S3261/.S2103/.Gzerl
1 3 3 1 3 3 2
Ga(-(-)) Ga(-) Ga(-(-)) Ga(-)
2 2 2 2 2
2 Ga(1) (--------------- - ----------------------)
3 1 3 3 1 2 2
Ga(-(-)) Ga(-) Ga(-(-)) Ga(-) Ga(1)
2 2 2 2
Out[33]= ---------------------------------------------------
3 2
3 Ga(-)
2
In[42]:= Simplify[R]
1 3 2 1 3
2 Ga(-(-)) (Ga(1) - Ga(-) Ga(-))
2 2 2
Out[42]= ----------------------------------
3 1 3 3
3 Ga(-(-)) Ga(-) Ga(-)
2 2 2
In[52]:= ?S3261
Summation formula (Slater, Appendix (III.31)) in form of a rule.
In[53]:= ?S2103
Summation formula (Slater, Appendix (III.3)) in form of a rule.
This is not a proof.
$$S_p=\sum\limits_{n=1}^{p}\frac{H_{n}^{(2)}}{n^2}$$ $$S_p=\frac{\left(H_p^{(2)}\right){}^2}{2}-\frac{H_p^{(4)}}{2}-\frac{\psi ^{(3)}(p+1)}{6}+\frac{\pi ^4}{90}$$ Expanded as series $$S_p=\frac{7 \pi ^4}{360}-\frac{\pi ^2}{6 p}+\frac{6+\pi ^2}{12 p^2}+O\left(\frac{1}{p^3}\right)$$
Now, if you consider $$T_k=\sum\limits_{n=1}^{\infty}\frac{H_{n}^{(2k)}}{n^{2k}}$$ you generate
$$T_k= a_k\, \zeta(4k)$$ where the coefficients are $$\left\{\frac{7}{4},\frac{13}{12},\frac{703}{691},\frac{14527}{144 68},\frac{524354}{523833},\frac{3546333857}{3545461365},\frac{ 6785975897}{6785560294},\frac{30837755428255}{30837284164868} \right\}$$
As @Notamathematician commented, the numerators correspon to sequence $A348830$ in $OEIS$. Sequence $A348829$ gives the difference between numerator and denominator.
Edit
Interesting is the odd case
$$U_k=\sum\limits_{n=1}^{\infty}\frac{H_{n}^{(2k+1)}}{n^{2k+1}}=\frac 12 \big(\zeta(2k+1)\big)^2+ \,\zeta(2(2k+1)\big) $$
Best Answer
Using the standard power series for the complete elliptic integral of the second kind $$E(k) = \frac{\pi}{2} \sum_{j=0}^\infty \left(\frac{(2j)!}{2^{2j}(j!)^2}\right)^2 \frac{k^{2j}}{1-2j},$$ we find \begin{align*} \sum\limits_{j=1}^\infty \frac{(2j-3)!!\,(2j-1)!!}{(2j-2)!!\,(2j+2)!!} k^{2j}&=\sum_{j=1}^\infty\frac{-j}{j+1} \left(\frac{(2j)!}{2^{2j}(j!)^2}\right)^2 \frac{k^{2j}}{1-2j} \\ &= -\frac{1}{k^2} \int_0^k\mathrm{d}k\,k^2 \frac{\mathrm{d}}{\mathrm{d}k}\left(\frac{2}{\pi}E(k)\right)\\ &= \frac{2}{3}\frac{k^2-1}{k^2}\frac{2}{\pi}K(k) - \frac{k^2-2}{3k^2}\frac{2}{\pi}E(k). \end{align*} In the limit $k\to 1$ only the second term survives with $E(1)=1$ and therefore \begin{align*} \sum\limits_{j=1}^\infty \frac{(2j-3)!!\,(2j-1)!!}{(2j-2)!!\,(2j+2)!!} &=\frac{2}{3\pi}. \end{align*}