Class Of Infinite Series Research Articles

Two Classes of Infinite Series through Contour Integration

Two Classes of Infinite Series through Contour Integration

Remarkable series concerning $ \binom{3n}{n} $ and harmonic numbers in numerators

<abstract><p>Three classes of infinite series containing binomial coefficient $ \binom{3n}{n} $, harmonic-like numbers, and an independent variable "$ y $" are examined. Several algebraic formulae in closed form are established, including, as special cases, three conjectured values for numerical series by Z.-W. Sun. This is fulfilled by integrating Lambert's series and manipulating the cubic transformations for the $ _3{F_2} $-series through the "coefficient extraction" method.</p></abstract>

Four Variants of Riemann Zeta Function

By means of the generating function method and Dougall’s formulae for bilateral hypergeometric series, we examine four classes of infinite series, which may be considered as variants of Riemann zeta function. Several summation formulae are established in closed form, which shows remarkably that the values of these series result in multiples of integer powers of π either by rational numbers or by algebraic numbers.

Some Evaluation of Infinite Series Involving Trigonometric and Hyperbolic Functions

In this paper, by using the residue theorem and asymptotic formulas of trigonometric and hyperbolic functions at the poles, we establish many relations involving two or more infinite series of trigonometric and hyperbolic functions. In particular, we evaluate in closed form certain classes of infinite series containing hyperbolic functions, which are related to Gamma functions and power of $$\pi $$ . Moreover, some interesting new consequences and illustrative examples are considered.

Some continued fractions for π and G

We present here two classes of infinite series and the associated continued fractions involving π and Catalan’s constant based on the work of Euler and Ramanujan. A few sundry continued fractions are also given.

The Riemann zeta function and classes of infinite series

We present one-parameter series representations for the following series involving the Riemann zeta function ??n=3 n odd ?(n)/n sn and ??n=2 n even ?(n) n sn and we apply our results to obtain new representations for some mathematical constants such as the Euler (or Euler-Mascheroni) constant, the Catalan constant, log 2, ?(3) and ?.

A UNIFIED METHOD FOR EVALUATING RIEMANN ZETA FUNCTIONS, DIRICHLET SERIES, ASSOCIATED CLAUSEN FUNCTIONS, OTHER ALLIED SERIES, AND NEW CLASSES OF INFINITE SERIES

Abstract: We have shown here for the first time that the completeness relation provides a simple unified theoretical framework for deriving different kinds of new recurrence formulae for Riemann Zeta Functions, Dirichlet series and Other Allied Series by selecting only different forms of complete set of orthonormal function (CSOF) in contrast to the expansion method (EM) where one needs to select not only different kinds of CSOF but also suitable arbitrary function. A new proof is also given by selecting only orthogonal Bessel functions for the well known identity corresponding to the sum of squares of the reciprocals of zeros for the m-th order Bessel function. In addition, we have shown here that, in comparison to the EM and other methods, our present method has far-reaching implications, viz. (i) All proofs are based on completeness relation and proper selection of orthogonal functions without selecting any arbitrary functions. (ii) Simpler proofs are possible for new identities corresponding to infinite number of infinite series, sum of each having a fixed value pi (π ). (iii) New proofs emerge not only for the identities corresponding to the Associated Clausen functions but also the sum of new classes of infinite series which resemble the associated Clausen functions.

The Summation of One Class of Infinite Series

This paper presents closed-form expressions for the series, , where the sum is from n = 1 to n = ∞. These expressions were obtained by recasting the series in a different form, followed by the use of certain relationships involving the elliptical nome. Among the values of x for which these expressions can be obtained are of the form: and , where l is an integer between –∞ and ∞. The values of λ include 1,,and 3. Examples of closed-form expressions obtained in this manner are first presented for , , , and . Additional examples are then presented for , , , and . This undertaking was prompted by the author’s work on an electrostatics boundary-value problem related to the van der Pauw measurement technique of electrical resistivity. The presence of this series for x = in the solution of that problem and its absence from any compendium of infinite series that he consulted led to this work.

INFINITE SERIES RELATION FROM A MODULAR TRANSFORMATION FORMULA FOR THE GENERALIZED EISENSTEIN SERIES

In 1970s, B. C. Berndt proved a transformation formula for a large class of functions that includes the classical Dedekind eta function. From this formula, he evaluated several classes of infinite series and found a lot of interesting infinite series identities. In this paper, using his formula, we find new infinite series identities.

Eisenstein series associated with Γ0(2)

In this paper, we define the normalized Eisenstein series ℘, e, and $\mathcal{Q}$ associated with Γ0(2), and derive three differential equations satisfied by them from some trigonometric identities. By using these three formulas, we define a differential equation depending on the weights of modular forms on Γ0(2) and then construct its modular solutions by using orthogonal polynomials and Gaussian hypergeometric series. We also construct a certain class of infinite series connected with the triangular numbers. Finally, we derive a combinatorial identity from a formula involving the triangular numbers.

Further summation formulae related to generalized harmonic numbers

By employing the univariate series expansion of classical hypergeometric series formulae, Shen [L.-C. Shen, Remarks on some integrals and series involving the Stirling numbers and ζ ( n ) , Trans. Amer. Math. Soc. 347 (1995) 1391–1399] and Choi and Srivastava [J. Choi, H.M. Srivastava, Certain classes of infinite series, Monatsh. Math. 127 (1999) 15–25; J. Choi, H.M. Srivastava, Explicit evaluation of Euler and related sums, Ramanujan J. 10 (2005) 51–70] investigated the evaluation of infinite series related to generalized harmonic numbers. More summation formulae have systematically been derived by Chu [W. Chu, Hypergeometric series and the Riemann Zeta function, Acta Arith. 82 (1997) 103–118], who developed fully this approach to the multivariate case. The present paper will explore the hypergeometric series method further and establish numerous summation formulae expressing infinite series related to generalized harmonic numbers in terms of the Riemann Zeta function ζ ( m ) with m = 5 , 6 , 7 , including several known ones as examples.

A New Identity for (q;q) 10 with an Application to Ramanujan's Partition Congruence Modulo 11

Our principal second goal is to show that this identity leads to a short proof of Ramanujan’s famous congruence p(11n+6) ≡ 0 (mod 11), where p(n) denotes the number of unrestricted partitions of the positive integer n. Our proof of Theorem 2.2 is short but depends upon some results of Ramanujan from his notebooks [18]. In Section 4, we give a more elementary proof, based upon Ramanujan’s 1ψ1 summation formula, of Ramanujan’s principal result, Lemma 2.1, which is employed in our proof of Theorem 2.2. In Section 5, we give an entirely different and direct proof of Theorem 2.2 based on several elementary identities for theta functions due to Ramanujan. In fact, during our first proof of Theorem 2.2, we establish a special case of a general result on Eisenstein series found in Ramanujan’s lost notebook [19, p. 369]. More precisely, Ramanujan asserts that every member of a certain class of infinite series can be expressed in terms of Ramanujan’s Eisenstein series P , Q, and R (to be defined in Section 6). A less precise version of this claim appears in Ramanujan’s notebooks [18], [2, p. 65, Entry 35(i)], but we prove the better version in Section 6. D. Stanton empirically discovered an analogue of Theorem 2.2, and in Section 7 we give a proof of Stanton’s identity. In Section 8, we prove that (q; q) ∞ is lacunary; in this connection, see a result of J.–P. Serre [20].

Classroom note: Evaluating a class of series using Taylor's theorem

A class of infinite series is evaluated with the aid of Taylor's theorem and a comparison is made with other methods.

Taylor's theorem and integral expansions- a class of infinite series

Taylor's theorem and integral expansions- a class of infinite series

Certain Classes of Infinite Series

The authors evaluate some interesting families of infinite series by analyzing known identities involving generalized hypergeometric series. Several special cases of the main results are shown to be related to earlier works on the subject.

A class of infinite series involving binomial and factorial coefficients∗

† This work was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant A‐73S3. Starting from an elementary summation formula involving binomial coefficients, D. Sullivan and J. C. F. Sullivan [1] derived the sums of a number of interesting series involving binomial and factorial coefficients. The object of this paper is to present a systematic analysis of these and numerous other analogous sums which are shown to be simple consequences of a well‐known result in the theory of the familiar hypergeometric series.

The irrationality of certain infinite series

The aim of this paper is to prove the irrationality of a certain class of infinite series. The main theorem is related to some results due to Erdos and Straus [7], Erdos [5] and Sándor [15]. As applications of the main result, the solutions of two problems posed by Erdös and Graham [6] are given, among others.

A necessary and sufficient condition for the uniform convergence of a certain class of infinite series

A necessary and sufficient condition for the uniform convergence of a certain class of infinite series

VIII. Investigation of formulæ for the summation of certain classes of infinite series

VIII. <i>Investigation of formulæ for the summation of certain classes of infinite series</i>

XXVI. Investigation of formulæ for the summation of certain classes of infinite series

XXVI. <i>Investigation of formulæ for the summation of certain classes of infinite series</i>