[seqfan] Re: Connection between A2450 and A321873?

[jean-paul allouche]

Post-scriptum :

The sum Sum_{k>=1} 3/(4^k - 1) appears in a paper of Kurokawa and Wakayama
entitled "ON q-ANALOGUES OF THE EULER CONSTANT AND LERCH’S LIMIT FORMULA"
in Proc. Amer. Math. Soc. 
(https://www.ams.org/journals/proc/2004-132-04/S0002-9939-03-07025-4/S0002-9939-03-07025-4.pdf)
where (see p. 937) one can find the equality Sum_{k>=1} 3/(4^k - 1) = 
gamma(4) + 3/2 - (3 log 3)/(2 log 2)
Also Theorem 2.2 in the same paper (p. 938) states that gamma(q) = - 
(Gamma_q)'(1) (q-1)/(log q),
where Gamma_q(x) is the q-Gamma function of Jackson (see def. bottom of 
p. 937).

I guess that Stier's formula is now a matter of knowing what 
"QPolyGamma" is and to relate it to (Gamma_q)'(1)

best wishes
jean-paul


Le 23/10/2019 à 09:41, jean-paul allouche a écrit :
> Hi
>
> Both sequences involve Sum_{k>=1} 3/(4^k - 1)
> Thus the only question is to justify the claim:
>
>
> Sum_{n>=1} 1/a(n) converges to 
> (3*(log[4/3]-QPolyGamma[0,1,1/4]))/log[4] =
> 1.26329305810027133188797266393139146884324008972340089723642183177... 
> -K. G. Stier <http://oeis.org/wiki/User:K._G._Stier>, Jun 23 2014
>
> Series similar to Sum_{k<=1} 3/(4^k-1) can be found, e.g., in
> Borwein, J. M. and Borwein, P. B. "Evaluation of Sums of Reciprocals
> of Fibonacci Sequences." §3.7 in Pi & the AGM: A Study in Analytic
> Number Theory and Computational Complexity. New York: Wiley, pp. 
> 91-101, 1987,
> where they are expressed in terms of a theta function. It remains to 
> prove
> the link with the PolyGamma function
>
> best wishes
> jp
>
>
>
> Le 23/10/2019 à 06:30, Alonso Del Arte a écrit :
>> One of the comments in A002450 states that
>>
>>> Sum_{n >= 1} 1/a(n) converges to (3*(log[4/3] - QPolyGamma[0, 1,
>> 1/4]))/log[4] = 1.263293...
>>
>> Is this number the constant in A321873?
>>
>> Al
>>
>
>
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