[Paper Review] Vacca-type series for values of the generalized-Euler-constant function and its derivative

Khodabakhsh Hessami Pilehrood, Tatiana Hessami Pilehrood|arXiv (Cornell University)|Aug 4, 2008

Advanced Mathematical Identities6 references3 citations

TL;DR

This paper generalizes Vacca-type and Ramanujan-type series for values of the generalized-Euler-constant function $\gamma_{a,b}(z)$ and its derivative using integral representations involving generating functions of digit counts in base-$B$ expansions. It derives new rational series for $\log\frac{4}{\pi}$, $\frac{G}{\pi}$, $\frac{\zeta'(2)}{\pi^2}$, and logarithms of Somos’s and Glaisher-Kinkelin’s constants, with explicit convergent series expansions based on base-$B$ digit structures.

ABSTRACT

We generalize well-known Catalan-type integrals for Euler's constant to values of the generalized-Euler-constant function and its derivatives. Using generating functions appeared in these integral representations we give new Vacca and Ramanujan-type series for values of the generalized-Euler-constant function and Addison-type series for values of the generalized-Euler-constant function and its derivative. As a consequence, we get base $B$ rational series for $\log\frac{4}π,$ $\frac{G}π$ (where $G$ is Catalan's constant), $\frac{ζ'(2)}{π^2}$ and also for logarithms of Somos's and Glaisher-Kinkelin's constants.

Motivation & Objective

• To extend Catalan-type integrals and Vacca-type series for Euler’s constant to the generalized-Euler-constant function $\gamma_{a,b}(z)$ and its derivative.
• To derive new rational series for special constants such as $\log\frac{4}{\pi}$, $\frac{G}{\pi}$, $\frac{\zeta'(2)}{\pi^2}$, and logarithms of Somos’s and Glaisher-Kinkelin’s constants.
• To establish connections between digit-counting functions in base-$B$ expansions and series summation via generating functions and integral transforms.
• To provide fast-converging expansions for $\gamma_{a,b}(z)$ and its derivative using polynomial-weighted sums over base-$B$ digit structures.

Proposed method

• Generalize Ramanujan and Berndt-Bowman integrals for $\gamma$ to the generalized-Euler-constant function $\gamma_{a,b}(z)$ using generating functions of digit counts in base-$B$ expansions.
• Define $L_B(k) = \lfloor \log_B(Bk) \rfloor$, the base-$B$ length of $k$, and use it as a weight in series expansions.
• Construct generating functions $G_B(x)$ from the number of occurrences of digits or words in base-$B$ expansions, which encode $N_{\omega,B}(n)$, the count of a word $\omega$ in $n$'s base-$B$ representation.
• Derive new Vacca-type series via integrals of the form $\int_0^1 \frac{1}{1+x} \sum_{n=1}^\infty x^{B^n - 1} dx$, extended to $\gamma_{a,b}(z)$.
• Use the polynomial $P_B(n)$ from Sondow’s generalized Addison series to weight terms in rational series for $\gamma_{a,b}(z)$ and its derivative.
• Apply recursive sequences $a_{k,\ell}$ and $c_k$ derived from generating functions to express coefficients in the final series expansions.

Experimental results

Research questions

• RQ1 Can Vacca-type and Ramanujan-type series for Euler’s constant be generalized to values of the generalized-Euler-constant function $\gamma_{a,b}(z)$ and its derivative?
• RQ2 What rational series representations can be derived for $\log\frac{4}{\pi}$, $\frac{G}{\pi}$, $\frac{\zeta'(2)}{\pi^2}$, and logarithms of special constants using base-$B$ digit structures?
• RQ3 How do generating functions of digit counts in base-$B$ expansions relate to integral representations of $\gamma_{a,b}(z)$?
• RQ4 Can fast-converging series for $\gamma_{a,b}(z)$ and $\gamma'_{a,b}(z)$ be constructed using weighted sums over base-$B$ digit counts?
• RQ5 What recursive sequences govern the coefficients in the new series expansions for special constants?

Key findings

• The paper derives a base-$B$ rational series for $\log\frac{4}{\pi}$: $\log\frac{4}{\pi} = \frac{5}{8} - \frac{1}{2}\sum_{k=1}^\infty \frac{a_k}{2k(2k+1)(2k+2)}$, where $a_1 = 4$ and $a_k = a_{\lfloor k/2 \rfloor} + \frac{1}{2^{k-1}}$ for $k \geq 2$.
• A new series for the Glaisher-Kinkelin constant is given by $\log A = \frac{13}{48} - \frac{1}{36}\sum_{k=1}^\infty \left(7L_B(k) - 7L_B(\lfloor k/2 \rfloor) + b_k\right)\frac{P_B(k)}{Bk(Bk+1)\cdots(Bk+B)}$, with $b_k$ recursively defined.
• For $\frac{G}{\pi}$, the paper establishes $\frac{G}{\pi} = \frac{11}{32} + \sum_{k=1}^\infty \left(\frac{1}{8}L_B(\lfloor k/2 \rfloor) - \frac{1}{8}L_B(k) + c_k\right)\frac{P_B(k)}{Bk(Bk+1)\cdots(Bk+B)}$, with $c_k$ defined recursively.
• The series for $\frac{\zeta'(2)}{\pi^2}$ is given by $\frac{\zeta'(2)}{\pi^2} = -\frac{1}{16} + \frac{1}{36}\sum_{k=1}^\infty \left(4L_B(k) - L_B(\lfloor k/2 \rfloor) + c_k\right)\frac{P_B(k)}{Bk(Bk+1)\cdots(Bk+B)}$, with $c_k$ recursively defined.
• The paper provides a new integral representation for $\gamma_{2,1}(-1)$ as $\frac{\pi}{8} - \frac{1}{4}\log 2 + \sum_{k=1}^\infty a_{k,0}\frac{P_B(k)}{Bk(Bk+1)\cdots(Bk+B)}$, where $a_{k,0}$ is recursively defined.
• For $\gamma'_{2,1}(-1)$, the paper gives $\gamma'_{2,1}(-1) = \frac{\pi}{16} - \frac{1}{4}\log 2 + \sum_{k=1}^\infty a_{k,1}\frac{P_B(k)}{Bk(Bk+1)\cdots(Bk+B)}$, with $a_{k,1}$ recursively defined.