scholarly journals Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions

2021 ◽  
Vol 6 (7) ◽  
pp. 7494-7517
Author(s):  
Bai-Ni Guo ◽  
◽  
Dongkyu Lim ◽  
Feng Qi ◽  
◽  
...  
Keyword(s):  
Series Expansions ◽  
Bell Polynomials ◽  
Special Values ◽  
Series Representations

scholarly journals Special values of the Bell polynomials of the second kind for some sequences and functions

2020 ◽  
Vol 491 (2) ◽  
pp. 124382
Author(s):  
Feng Qi ◽  
Da-Wei Niu ◽  
Dongkyu Lim ◽  
Yong-Hong Yao
Keyword(s):  
Bell Polynomials ◽  
Special Values

scholarly journals Power series expansions of modular forms and p-adic interpolation of the square roots of Rankin–Selberg special values

2021 ◽  
pp. 1-41
Author(s):  
Andrea Mori
Keyword(s):  
Power Series ◽  
Modular Forms ◽  
Level Structure ◽  
Automorphic Representation ◽  
Base Change ◽  
Series Expansions ◽  
Square Roots ◽  
Infinite Type ◽  
Special Values ◽  
Power Series Expansions

Let [Formula: see text] be a newform of even weight [Formula: see text] for [Formula: see text], where [Formula: see text] is a possibly split indefinite quaternion algebra over [Formula: see text]. Let [Formula: see text] be a quadratic imaginary field splitting [Formula: see text] and [Formula: see text] an odd prime split in [Formula: see text]. We extend our theory of [Formula: see text]-adic measures attached to the power series expansions of [Formula: see text] around the Galois orbit of the CM point corresponding to an embedding [Formula: see text] to forms with any nebentypus and to [Formula: see text] dividing the level of [Formula: see text]. For the latter we restrict our considerations to CM points corresponding to test objects endowed with an arithmetic [Formula: see text]-level structure. Also, we restrict these [Formula: see text]-adic measures to [Formula: see text] and compute the corresponding Euler factor in the formula for the [Formula: see text]-adic interpolation of the “square roots”of the Rankin–Selberg special values [Formula: see text], where [Formula: see text] is the base change to [Formula: see text] of the automorphic representation of [Formula: see text] associated, up to Jacquet-Langland correspondence, to [Formula: see text] and [Formula: see text] is a compatible family of grössencharacters of [Formula: see text] with infinite type [Formula: see text].

Evaluation of the Wavelike Disturbance in the Kelvin Wave Source Potential

1988 ◽  
Vol 32 (01) ◽  
pp. 44-53 ◽  
Author(s):  
J. J. M. Baar ◽  
W. G. Price
Keyword(s):  
Numerical Evaluation ◽  
Field Component ◽  
Near Field ◽  
Neumann Series ◽  
Series Expansions ◽  
Effective Solution ◽  
Kelvin Wave ◽  
Wave Source ◽  
Series Representations ◽  
Single Integral

This paper discusses the numerical evaluation of the characteristic Kelvin wavelike disturbance trailing downstream from a translating submerged source. Mathematically the function describing the wavelike disturbance is expressed as a single integral with infinite integration limits and a rapidly oscillatory integrand. Numerical integration of such integrals is both cumbersome and time-consuming. Attention is therefore focused on two complementary Neumann-series expansions which were originally derived by Bessho [1].2 Numerically stable algorithms are presented for the accurate and efficient evaluation of the two series representations. When used in combination with the Chebyshev expansions for the nonoscillatory near-field component which were recently obtained by Newman [2], the present algorithms provide an effective solution to the numerical difficulties associated with the evaluation of the Kelvin wave source potential.

scholarly journals Fourier series of functions involving higher-order ordered Bell polynomials

2017 ◽  
Vol 15 (1) ◽  
pp. 1606-1617 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Gwan-Woo Jang ◽  
Lee Chae Jang
Keyword(s):  
Number Theory ◽  
Fourier Series ◽  
Higher Order ◽  
Series Expansions ◽  
Bell Polynomials ◽  
Bell Numbers ◽  
Enumerative Combinatorics ◽  
Bernoulli Functions ◽  
Series Of Functions

AbstractIn 1859, Cayley introduced the ordered Bell numbers which have been used in many problems in number theory and enumerative combinatorics. The ordered Bell polynomials were defined as a natural companion to the ordered Bell numbers (also known as the preferred arrangement numbers). In this paper, we study Fourier series of functions related to higher-order ordered Bell polynomials and derive their Fourier series expansions. In addition, we express each of them in terms of Bernoulli functions.

scholarly journals Series Representations at Special Values of Generalized Hurwitz-Lerch Zeta Function

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
S. Gaboury ◽  
A. Bayad
Keyword(s):  
Zeta Function ◽  
Higher Order ◽  
Euler Polynomials ◽  
Expansion Formula ◽  
Special Values ◽  
Lerch Zeta Function ◽  
Series Representations ◽  
Rational Arguments

By making use of some explicit relationships between the Apostol-Bernoulli, Apostol-Euler, Apostol-Genocchi, and Apostol-Frobenius-Euler polynomials of higher order and the generalized Hurwitz-Lerch zeta function as well as a new expansion formula for the generalized Hurwitz-Lerch zeta function obtained recently by Gaboury and Bayad , in this paper we present some series representations for these polynomials at rational arguments. These results provide extensions of those obtained by Apostol (1951) and by Srivastava (2000).

scholarly journals Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine functions and applications

2021 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Cosine Function ◽  
Binomial Coefficients ◽  
Series Expansions ◽  
Bell Polynomials ◽  
Specific Sequence ◽  
Hyperbolic Cosine ◽  
Faà Di Bruno Formula ◽  
Hyperbolic Cosine Function

Abstract In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for special values of the Bell polynomials of the second kind with respect to a specific sequence, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes Maclaurin's series expansions for real powers of the inverse cosine function and the inverse hyperbolic cosine function. By applying different series expansions for the square of the inverse cosine function, the author not only finds infinite series representations of the circular constant Pi and its square, but also derives two combinatorial identities involving central binomial coefficients.

scholarly journals Closed formulas for special bell polynomials by Stirling numbers and associate Stirling numbers

2020 ◽  
Vol 108 (122) ◽  
pp. 131-136
Author(s):  
Feng Qi ◽  
Dongkyu Lim
Keyword(s):  
Explicit Formula ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Special Values

We derive two explicit formulas for two sequences of special values of the Bell polynomials of the second kind in terms of associate Stirling numbers of the second kind, give an explicit formula for associate Stirling numbers of the second kind in terms of the Stirling numbers of the second kind, and, consequently, present two explicit formulas for two sequences of special values of the Bell polynomials of the second kind in terms of the Stirling numbers of the second kind.

scholarly journals Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine and sine functions with applications

2021 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Binomial Coefficients ◽  
Series Expansions ◽  
Bell Polynomials ◽  
Sine Function ◽  
Specific Sequence ◽  
Hyperbolic Cosine ◽  
Faà Di Bruno Formula ◽  
Sine Functions

Abstract In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for special values of the Bell polynomials of the second kind with respect to a specific sequence, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes Maclaurin's series expansions for real powers of the inverse cosine (sine) function and the inverse hyperbolic cosine (sine) function. By applying different series expansions for the square of the inverse cosine function, the author not only finds infinite series representations of the circular constant Pi and its square, but also derives two combinatorial identities involving central binomial coefficients.

scholarly journals Notes on two kinds of special values for the Bell polynomials of the second kind

2019 ◽  
Vol 20 (1) ◽  
pp. 465 ◽  
Author(s):  
Feng Qi ◽  
Dongkyu Lim ◽  
Yong-Hong Yao
Keyword(s):  
Bell Polynomials ◽  
Special Values
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