In this paper, we show how Zeilberger's algorithm can be used to find a closed-form expression for the sum of series containing a parameter. Series of the following form: (infinity)& sum;(k=0)(1/2-s)k(1/2+s)kk!(2 )1/(a +/- k+n)i and (infinity)& sum;(k=0)(1/2-s)k(1/2+s)kk!2 1/((a +/- k)n)i with i=1, 2, 3 and with(c)(n) in the Pochhammer notation, provide a natural extension of a family of series that Ramanujan expresse dusing the Landau constants. They are ideally suited for applying Zeilberger's algorithm. Zeilberger's algorithm finds a linear recurrence for a sum of hypergeometric terms. By choosing these terms wisely, the resulting recurrence can sometimes be solved in closed form. Asan example, we give a solution to an open problem in a 2022 article by Stewart that uses a series of this type.

The Basel problem is known to be one of those mathematical problems with the most published proofs. In this paper, we offer one proof more, a computer-assisted one, based on Zeilberger's algorithm, a method to find a recurrence relation for a sum of hypergeometric terms. The same algorithm is used to derive the well-known Madhava-Gregory-Leibniz series and some series found by Ramanujan from the Wallis product formula for pi.

The formula $${2\over \pi} = 1 - 5 \cdot \left({1\over 2} \right)<^>{3} + 9 \cdot \left({1 \cdot 3\over 2 \cdot 4} \right)<^>{3} - \cdots$$2 pi=1-5 & sdot;(12)3+9 & sdot;(1 & sdot;32 & sdot;4)3-& ctdot; was famously included as a discovery in Ramanujan's first letter to Hardy in 1913, and has been referred to as the Bauer-Ramanujan formula, in view of Bauer's 1859 proof of the above formula. There is a rich history associated with this formula and its many and dramatically different proofs, including a computer-based proof due to Zeilberger that may be seen as groundbreaking in the history of computer-assisted proofs. In addition to a complete survey we provide of all known proofs of the Bauer-Ramanujan formula, we introduce historical analyses based on these proofs, by arguing that the history of the Bauer-Ramanujan formula and our account of this history may be seen as being representative of much broader trends in the history of mathematics. In this regard, the earlier proofs tend to rely on one of the oldest and most basic tools in classical analysis, namely, interchanging the order of limiting operations. In contrast, the more modern proofs tend to rely on computer-related approaches toward summation problems, as in with Zeilberger-type and Gosper-type telescoping arguments.

Included in Ramanujan's first letter to Hardy was the remarkable formula 1 -5 (1/2 )(5) + 9(1 3/2 4)(5) - 13(1 3 5/ 2 4 6)(5) + Gamma(4)(1/4)/2 pi(4) . Different proofs of this formula have been given by a number of different authors, and this includes a recent proof due to Cantarini related to the generalized Clebsch-Gordan integral. Series involving fifth powers of binomial coefficients are known to be very difficult to evaluate, and it is not clear how to generalize the above formula. We introduce an evaluation technique based on a Fourier-Legendre expansion that was considered by Baranov in 2006, and we succeed in applying our technique to obtain families of generalizations and variants of Ramanujan's formula.

the following question arises: Can this latter formula be derived by squaring both sides of the former? There have been several proofs of Euler's formula, or its equivalent formulation ζ(2)=π2/6, based on the idea of squaring 1−13+15−⋯=π4, including a proof presented in a letter from Euler to Goldbach dating from 1742. We consider the history of proofs of this form, and we offer another simple proof of ζ(2)=π2/6 that also relies on squaring Gregory's series.

We propose a relation between values of the Riemann zeta function zeta and a family of integrals. This results in an integral representation for zeta(2p), where p is a positive integer, and an expression of zeta(2p + 1) involving one of the above-mentioned integrals together with a harmonic-number sum. Simplification of the latter eventually leads to an integral representation of zeta(2p + 1).

Let $ F(n,k) $ F(n,k) be a hypergeometric function that may be expressed so that n appears within initial arguments of inverted Pochhammer symbols, as in factors of the form $ \frac {1}{(n)_{k}} $ 1(n)k. Only in exceptional cases is $ F(n, k) $ F(n,k) such that Zeilberger's algorithm produces a two-term recursion for $ \sum _{k = 0}<^>{\infty } F(n, k) $ n-ary sumation k=0 infinity F(n,k) obtained via the telescoping of the right-hand side of a difference equation of the form $ p_{1}(n) F(n + r, k) + p_{2}(n) F(n, k) = G(n, k+1) - G(n, k) $ p1(n)F(n+r,k)+p2(n)F(n,k)=G(n,k+1)-G(n,k) for fixed $ r \in \mathbb {N} $ r is an element of N and polynomials $ p_{1} $ p1 and $ p_{2} $ p2. Building on the work of Wilf, we apply a series acceleration technique based on two-term hypergeometric recursions derived via Zeilberger's algorithm. Fast converging series previously given by Ramanujan, Guillera, Chu and Zhang, Chu, Lupas, and Amdeberhan are special cases of hypergeometric transforms introduced in our article.

In 2002 and 2006, using a Wilf- Zeilberger-based method, Guillera introduced proofs for evaluations for what are considered as the simplest two series out of Ramanujan's 17 series for 1/pi. In this article, we show how the WZ method may be used in a fundamentally and nontrivially different way to prove these results, and to obtain identities for infinite families of Ramanujan-like series for 1/pi. We introduce a F-3(2)-recurrence that we had discovered experimentally, and we prove this recursion using the WZ method and apply it to obtain a series acceleration formula that we apply to formulate a new and simple proof for the Ramanujan series for 1/pi that has a convergence rate of 1/64, and we provide an infinite family of generalizations of this formula, and similarly for Ramanujan's series of convergence rate 1/4.

We examine the structure of the periodic continued fractions of square roots of non-square positive integers given by an integer-valued quadratic polynomial Q(n) = (a(n) + b)(2) + (gn + h). The aim is to identify repeated blocks of partial quotients in the period. The quotients in the period form a palindrome, and when the period length is even, the period has a central term an. The paper focuses on periods with a(n) = a(0) or a(n) = a(0 - 1), where a(0) is the initial partial quotient. For a(n) = a(0) we give an algorithm to obtain formulas involving repeated blocks comprising three or more elements, not all equal.

We prove a conjecture due to Chu concerning Gosper-type sums, using an evaluation due to Chudnovsky and Chudnovsky in 1998. This formula discovered by the Chudnovsky brothers was later rediscovered by Borwein and Girgensohn, but no proof of this formula has been given, prior to our article. We introduce a full, self-contained proof of the Chudnovsky–Chudnovsky evaluation, to formulate a full solution to a problem due to Chu on Gosper-type sums involving reciprocals of binomial coefficients of the form ( [ 3n + ε; n ]) for n ∈ℕ_0 and ε∈{ 1, ± 2 } .

We introduce a full solution to a problem considered by Wang and Chu concerning series involving the squares of finite sums of the form 1 + 1/3 + ⋯ + 1/2n-1 . Our proof involves techniques from the theory of colored multiple zeta values.

In 2010, Kh. Hessami Pilehrood and T. Hessami Pilehrood introduced generating function identities used to obtain series accelerations for values of Dirichlet's $\beta$ function, via the Markov--Wilf--Zeilberger method. Inspired by these past results, together with related results introduced by Chu et al., we introduce a variety of hypergeometric recurrences. We prove these recurrences using the WZ method, and we apply these recurrences to obtain series acceleration identities. We introduce a family of summations generalizing a Ramanujan-type series for $\frac{1}{\pi^2}$ due to Guillera, and a family of summations generalizing an accelerated series for Catalan's constant due to Lupa\c{s}, and many related results.

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We consider here a trilogarithmic expression that plays a similar role to Catalan's constant G in many ways:The main purpose of this article is to demonstrate how G is a naturally occurring and useful expression that deserves to be recognized as a mathematical constant and as a natural trilogarithmic "extension" of Catalan's constant G. Having identified this constant, we evaluate many new and non-trivial integrals, Euler-type sums, p F q series, and binomial-harmonic series using G , extending known results on the classical version of Catalan's constant.

With some thought and imagination, it is possible to create interesting and accessible problems for undergraduate students using the area and volume applications considered here. If nothing else, the examples presented in this Note provide some geometric motivation for power means associated with two positive numbers. We should mention that power means are defined for finite sets of non-negative numbers; see [1] for an extensive discussion of the properties of such means. (An Internet search also reveals a great deal of information about these generalised means.)