Related papers: Series for $1/\pi$ Using Legendre's Relation
In our recent works [R. Szmytkowski, J. Phys. A 39 (2006) 15147; corrigendum: 40 (2007) 7819; addendum: 40 (2007) 14887], we have investigated the derivative of the Legendre function of the first kind, $P_{\nu}(z)$, with respect to its…
Re presenting the traditional proof of Srinivasa Ramanujan's own favorite series for the reciprocal of $\pi$ :\begin{equation}\frac{1}{\pi} = \frac{\sqrt{8}}{9801} \sum_{n=0}^{+\infty} \frac{(4n)!}{(n!)^4} \frac{1103 + 26390n}{396^{4n}} \;…
The $p$-th power of the logarithm of the Catalan generating function is computed using the Stirling cycle numbers. Instead of Stirling numbers, one may write this generating function in terms of higher order harmonic numbers.
The author gives the full list of his conjectures on series for powers of $\pi$ and other important constants scattered in some of his public papers or his private diaries. The list contains 234 reasonable conjectural series. On the list…
Discrete analogs of the classical Mehler-Fock transforms are introduced and investigated. It involves series with the associated Legendre function $P^\mu_{in-1/2}(x), x > 1,\ {\rm Re} \mu < 1/2, \ n \in \mathbb{N}, i $ is the imaginary…
We describe a method for the numerical evaluation of normalized versions of the associated Legendre functions $P_\nu^{-\mu}$ and $Q_\nu^{-\mu}$ of degrees $0 \leq \nu \leq 1,000,000$ and orders $-\nu \leq \mu \leq \nu$ on the interval…
Power series are introduced that are simultaneously convergent for all real and p-adic numbers. Our expansions are in some aspects similar to those of exponential, trigonometric, and hyperbolic functions. Starting from these series and…
The author's method (math-ph/9804010) that uses the Laplace transform to find exact values for a large class of convergent series is extended to trigonometric series.
We have constructed new formulae for generation of solutions for the nonlinear heat equation and for the Burgers equation that are based on linearizing nonlocal transformations and on nonlocal symmetries of linear equations. Found nonlocal…
We translate the concept of succession rule and the ECO method into matrix notation, introducing the concept of a production matrix. This allows us to combine our method with other enumeration techniques using matrices, such as the method…
We propose and investigate a bi-infinite matrix approach to the multiplication and composition of formal Laurent series. We generalize the concept of Riordan matrix to this bi-infinite context, obtaining matrices that are not necessarily…
In this note we augment the poly-Bernoulli family with two new combinatorial objects. We derive formulas for the relatives of the poly-Bernoulli numbers using the appropriate variations of combinatorial interpretations. Our goal is to show…
We derive 10 new Ramanujan-Sato series of $1/\pi$ by using the method of Huber, Schultz and Ye. The levels of these series are 14, 15, 16, 20, 21, 22, 26, 35, 39.
Using a pointwise version of Fej\'{e}r's theorem about Fourier series, we obtain two formulae related to the series representations of positive integral powers of $\pi$. We also check the correctness of our formulae by the applications of…
We develop a differential theory for the polarity transform parallel to that for the Legendre transform, which is applicable when the functions studied are "geometric convex", namely convex, non-negative and vanish at the origin. This…
Approximate relations between $e$ and $\pi$ are reviewed, some new connections being established. Nilakantha's series expansion for $\pi$ is transformed to accelerate its convergence. Its comparison with the standard inverse-factorial…
Expressions for the derivatives with respect to order of modified Bessel functions evaluated at integer orders and certain integral representations of associated Legendre functions with modulus argument greater than unity are used to…
We recast homogeneous linear recurrence sequences with fixed coefficients in terms of partial Bell polynomials, and use their properties to obtain various combinatorial identities and multifold convolution formulas. Our approach relies on a…
We report results on various techniques which allow to compute the expansion into Legendre (or in general Gegenbauer) polynomials in an efficient way. We describe in some detail the algebraic/symbolic approach already presented in Ref.1 and…
In an unpublished 2005 paper T. Rivoal proved a formula giving 4/pi as the infinite product of factors (1 + 1/(k+1)) to a power involving the integer part of the logarithm of k in base 2 and a 4-periodic sequence. We show how a lemma in a…