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"Infinite series"
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FINSLERIAN HYPERSURFACES OF A FINSLER SPACE WITH DEFORMED DOUGLAS INFINITE SERIES METRIC
In present paper we studied the geometrical properties of Finslerian hy-persurfaces and its reducibility of Cartan C- tensor in various forms for a Finsier space Fn equipped with defbrmed Infinite series metric. Fürther we obtained the value of main scalar I for the hypersurface in a two-dimensional case.
Journal Article
Gauss’ Second Theorem for F12(1/2)-Series and Novel Harmonic Series Identities
Two summation theorems concerning the F12(1/2)-series due to Gauss and Bailey will be examined by employing the “coefficient extraction method”. Forty infinite series concerning harmonic numbers and binomial/multinomial coefficients will be evaluated in closed form, including eight conjectured ones made by Z.-W. Sun. The presented comprehensive coverage for the harmonic series of convergence rate “1/2” may serve as a reference source for readers.
Journal Article
Power Series Expansions of Real Powers of Inverse Cosine and Sine Functions, Closed-Form Formulas of Partial Bell Polynomials at Specific Arguments, and Series Representations of Real Powers of Circular Constant
In this paper, by means of the Faà di Bruno formula, with the help of explicit formulas for partial Bell polynomials at specific arguments of two specific sequences generated by derivatives at the origin of the inverse sine and inverse cosine functions, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes power series expansions for real powers of the inverse cosine (sine) functions and the inverse hyperbolic cosine (sine) functions. By comparing different series expansions for the square of the inverse cosine function and for the positive integer power of the inverse sine function, the author not only finds infinite series representations of the circular constant π and its real powers, but also derives several combinatorial identities involving central binomial coefficients and the Stirling numbers of the first kind.
Journal Article
Balancing Polynomials, Fibonacci Numbers and Some New Series for π
We evaluate some types of infinite series with balancing and Lucas-balancing polynomials in closed form. These evaluations will lead to some new curious series for
π
involving Fibonacci and Lucas numbers. Our findings complement those of Castellanos from 1986 to 1989.
Journal Article
Infinite Series Concerning Tails of Riemann Zeta Values
Infinite series involving Riemann’s zeta and Dirichlet’s lambda tails, and weighted by three harmonic-like elementary symmetric functions are examined. By means of integral representations of zeta tails together with the telescopic approach, twelve general summation theorems are established that express these series as coefficients of the bivariate beta function Beta(u,v). By further expanding Beta(u,v) into Laurent series in u and v, several explicit summation formulae are shown as consequences.
Journal Article
Modified Black Hole Potentials and Their Korteweg-de Vries Integrals: Instabilities and Beyond
Black Hole (BH) Quasi-Normal Modes (QNMs) and Greybody Factors (GBFs) are key signatures of BH dynamics that are crucial for testing fundamental physics via gravitational waves. Recent studies of the BH pseudospectrum have revealed instabilities in QNMs. Here, we introduce a new perspective using hidden symmetries in the BH dynamics, specifically the Korteweg-de Vries (KdV) integrals—an infinite series of conserved quantities. By analyzing modified BH potentials, we find strong evidence that KdV integrals are valuable indicators for detecting instabilities in QNMs and GBFs.
Journal Article
Construction of Infinite Series Exact Solitary Wave Solution of the KPI Equation via an Auxiliary Equation Method
The KPI equation is one of most well-known nonlinear evolution equations, which was first used to described two-dimensional shallow water wavs. Recently, it has found important applications in fluid mechanics, plasma ion acoustic waves, nonlinear optics, and other fields. In the process of studying these topics, it is very important to obtain the exact solutions of the KPI equation. In this paper, a general Riccati equation is treated as an auxiliary equation, which is solved to obtain many new types of solutions through several different function transformations. We solve the KPI equation using this general Riccati equation, and construct ten sets of the infinite series exact solitary wave solution of the KPI equation. The results show that this method is simple and effective for the construction of infinite series solutions of nonlinear evolution models.
Journal Article
Analysis of algebraic and geometric ideas in infinite series
As an important concept in mathematics, the convergence judgment of series has always been one of the research hotspots. As the reference series to judge the convergence and divergence of series, the proof of convergence of P-series and harmonic series has always been a research hotspot. This paper discusses the embodiment of algebra and geometry in the proof of convergence and divergence of P-series and harmonic series. Through the mutual transformation of geometry and algebra, the convergence and divergence of infinite series are analyzed, and the convergence problem of series is transformed into an intuitive geometric image for analysis. By introducing geometric intuition, it can not only simplify the solution process of the problem, but also help students and researchers better understand the essential characteristics of series, so as to provide new perspectives and ideas for the study of series convergence.
Journal Article
Infinite Series and Logarithmic Integrals Associated to Differentiation with Respect to Parameters of the Whittaker Mκ,μ(x) Function I
In this paper, first derivatives of the Whittaker function Mκ,μx are calculated with respect to the parameters. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of quotients of the digamma and gamma functions. Moreover, from the integral representation of Mκ,μx it is possible to obtain these parameter derivatives in terms of finite and infinite integrals with integrands containing elementary functions (products of algebraic, exponential, and logarithmic functions). These infinite sums and integrals can be expressed in closed form for particular values of the parameters. For this purpose, we have obtained the parameter derivative of the incomplete gamma function in closed form. As an application, reduction formulas for parameter derivatives of the confluent hypergeometric function are derived, along with finite and infinite integrals containing products of algebraic, exponential, logarithmic, and Bessel functions. Finally, reduction formulas for the Whittaker functions Mκ,μx and integral Whittaker functions Miκ,μx and miκ,μx are calculated.
Journal Article