arXiv subjects
Publications and source records attributed to Kunle Adegoke.
By utilizing two hypergeometric summation identities we extend two well-known Ramanujan series for $1/π$ by making each of them a member of an infinite family of series. We also find the corresponding families involving harmonic numbers and odd harmonic numbers.To further illustrate our method we derive a family of Ramanujan-like series associated with a hypergeometric series derived by Lavoie.
We develop a method for evaluating finite and infinite series involving Catalan numbers by specializing alternating binomial identities at half-integer parameters. The key ingredient is an explicit representation of the generalized binomial coefficients $\binom{m+\frac12}{k+1}$ in terms of Catalan numbers and products of odd linear factors. Combining this representation with Pascal-type identities and two identities of Batır and Sofo, we derive several families of closed-form summation formulas involving Catalan numbers, harmonic numbers, and odd harmonic numbers. As special cases, we obtain evaluations of finite sums together with their corresponding infinite-series analogues. The approach provides a systematic mechanism for transforming identities for generalized binomial coefficients into identities for Catalan-number series.
We develop a unified method for transforming and evaluating infinite series involving products of terms of second-order linear recurrences in the denominator. The approach is based on a discrete Abel-type summation formula (summation by parts), which converts reciprocal series with three or more factors into expressions exhibiting a partial telescoping structure. As a consequence, we obtain general transformation formulas for series of the form $\sum\limits_{k=1}^{\infty} \frac{(\pm 1)^k}{w_{rk+l} w_{mk+s} w_{m(k+1)+s}}$, together with extensions to products of four or more terms. These formulas provide a systematic framework that unifies and extends many known identities for Fibonacci and Lucas numbers. In addition, the method leads to explicit evaluations and identities involving several classical combinatorial sequences, including Catalan numbers, harmonic numbers, and Stirling numbers of both kinds. A key feature of the approach is that it naturally distinguishes between even and odd values of the parameter $m$, leading to structurally different representations. The results show that summation by parts is an effective and flexible tool for reducing multi-factor reciprocal sums to simpler forms.
Using generalized binomial coefficient identities and some results of John Dougall, we derive some families of series involving the cubes of Catalan numbers. We also establish a family of series containing fourth powers of Catalan numbers. Finally, we find a generalization of the Bauer series for $1/π$ and obtain some Ramanujan-like series for $1/π^2$ and~$1/π^3$.
Starting with a known polynomial identity, we derive two generalizations of Touchard's identity concerning Catalan numbers; one obtained using the Beta function and the other via a connection with Stirling numbers of the second kind. We subsequently establish several new combinatorial identities.
Based on an interesting identity of Batır we derive new identities for double sums involving famous number sequences. We also prove some double sum identities for binomial transform pairs.
Given any two sequences of complex numbers, we establish simple relations between their binomial convolution and the binomial convolution of their individual binomial transforms. We employ these relations to derive new identities involving Fibonacci numbers, Bernoulli numbers, Catalan numbers, harmonic numbers, odd harmonic numbers, Stirling numbers of the second kind, and binomial coefficients. In addition, we present several results which allow the construction of new binomial-transform pairs from existing ones. Many new relations concerning self-inverse sequences are also derived.
In this paper, we continue our investigation of double sums where the inner sum is binomial but incomplete. We prove many new results for these types of double sums associated with binomial transform pairs. As applications we deduce new identities for double sums involving special numbers like Bernoulli numbers, Fibonacci numbers, harmonic numbers, Catalan numbers and Stirling numbers of the second kind. We also consider families of polynomials like Fibonacci polynomials, Chebyshev polynomials, Bernoulli polynomials, and others. Finally, we state new double sums involving hyperbolic functions.
In this paper, we find an elementary approach for double sums where the inner sum is binomial but incomplete. We apply our core identity and its relatives to double sums involving famous numbers such as harmonic numbers, Fibonacci numbers, Stirling numbers and $r$-Stirling numbers of the second kind.
We offer several new summation identities involving harmonic numbers, odd harmonic numbers, and Fibonacci numbers. Our results are derived using three different approaches: partial summation, polynomial identities and binomial transformation.
We give a short proof of the well-known Knuth's old sum and provide some generalizations. Our approach utilizes the binomial theorem and integration formulas derived using the Beta function. Several new polynomial identities and combinatorial identities are derived.
Using an elementary approach involving the Euler Beta function and the binomial theorem, we derive two polynomial identities; one of which is a generalization of a known polynomial identity. Two well-known combinatorial identities, namely Frisch's identity and Klamkin's identity, appear as immediate consequences of the polynomial identities. We subsequently establish several combinatorial identities, including a generalization of each of Frisch's identity and Klamkin's identity. Finally, we develop a scheme for deriving combinatorial identities associated with polynomial identities of a certain type.
In this paper, we explore a variety of series involving the central binomial coefficients, highlighting their structural properties and connections to other mathematical objects. Specifically, we derive new closed-form representations and examine the convergence properties of infinite series with a repeating alternation pattern of signs involving central binomial coefficients. More concretely, we derive the series $$\sum\limits_{n=0}^{\infty}\frac{(-1)^{ω_n}}{2n+1}\tbinom{2n}{n}x^n,\,\,\, \sum\limits_{n=0}^{\infty}{(-1)^{ω_n}}\tbinom{2n}{n}x^n\,\,\, \text{and} \,\,\, \sum\limits_{n=0}^{\infty}{(-1)^{ω_n}}n\tbinom{2n}{n}x^n,$$ where $ω_n$ represents both $\lfloor\frac{n}{2}\rfloor$ and $\lceil\frac{n}{2}\rceil$. Also, we present novel series involving Fibonacci and Lucas numbers, deriving many interesting identities.
We study three classes of combinatorial sums involving central binomial coefficients and harmonic numbers, odd harmonic numbers, and even indexed harmonic numbers, respectively. In each case we use summation by parts to derive recursive expressions for these sums. In addition, we offer an alternative approach to express one class of sums and some related sums in closed form in terms of Stirling numbers and r-Stirling numbers of the second kind.
In this paper, we present a general framework for the derivation of interesting finite combinatorial sums starting with certain classes of polynomial identities. The sums that can be derived involve products of binomial coefficients and also harmonic numbers and squared harmonic numbers. We apply the framework to discuss combinatorial sums associated with some prominent polynomial identities from the recent past.
Two classes of infinite series involving harmonic numbers and the binomial coefficient $C(3n,n)$ are evaluated in closed form using integrals. Several remarkable integral values and difficult series identities are stated as special cases of the main results.
Based on a recent representation of the psi function due to Guillera and Sondow and independently Boyadzhiev, new closed forms for various series involving harmonic numbers and inverse factorials are derived. A high point of the presentation is the rediscovery, by much simpler means, of a famous quadratic Euler sum originally discovered in 1995 by Borwein and Borwein.
We apply a seemingly forgotten series expression of Nörlund for the psi function to express infinite series involving inverse factorials in closed form. Many of such series contain products of Catalan numbers and (odd) harmonic numbers. We also prove some new series for $π$.