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Journal articles on the topic 'A Pochhammer symbol in Applied Mathematics'

Author: Grafiati

Published: 7 June 2025

Last updated: 12 July 2025

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1

Rahman, Gauhar, Abdus Saboor, Zunaira Anjum, Kottakkaran Sooppy Nisar, and Thabet Abdeljawad. "An Extension of the Mittag-Leffler Function and Its Associated Properties." Advances in Mathematical Physics 2020 (September 22, 2020): 1–8. http://dx.doi.org/10.1155/2020/5792853.

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Inspired by certain fascinating ongoing extensions of the special functions such as an extension of the Pochhammer symbol and generalized hypergeometric function, we present a new extension of the generalized Mittag-Leffler (ML) function εa,b;p,vκz1 in terms of the generalized Pochhammer symbol. We then deliberately find certain various properties and integral transformations of the said function εa,b;p,vκz1. Some particular cases and outcomes of the main results are also established.

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2

Masjed-Jamei, Mohammad, and Gradimir Milovanovic. "An extension of Pochhammer’s symbol and its application to hypergeometric functions, II." Filomat 32, no. 19 (2018): 6505–17. http://dx.doi.org/10.2298/fil1819505m.

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Recently we have introduced a productive form of gamma and beta functions and applied them for generalized hypergeometric series [Filomat, 31 (2017), 207-215]. In this paper, we define an additive form of gamma and beta functions and study some of their general properties in order to obtain a new extension of the Pochhammer symbol. We then apply the new symbol for introducing two different types of generalized hypergeometric functions. In other words, based on the defined additive beta function, we first introduce an extension of Gauss and confluent hypergeometric series and then, based on two additive types of the Pochhammer symbol, we introduce two extensions of generalized hypergeometric functions of any arbitrary order. The convergence of each series is studied separately and some illustrative examples are given in the sequel.

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3

You, Xu, Shou ou Huang, and Lu Liu. "Some inequalities associated to the ratio of Pochhammer k-symbol." Mathematical Inequalities & Applications, no. 1 (2017): 105–10. http://dx.doi.org/10.7153/mia-20-07.

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4

Niyaz, Mohamed, Ahmed H. Soliman, and Ahmed Bakhet. "Investigation of Fractional Calculus for Extended Wright Hypergeometric Matrix Functions." Abstract and Applied Analysis 2023 (April 28, 2023): 1–8. http://dx.doi.org/10.1155/2023/9505980.

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Throughout this paper, we will present a new extension of the Wright hypergeometric matrix function by employing the extended Pochhammer matrix symbol. First, we present the extended hypergeometric matrix function and express certain integral equations and differential formulae concerning it. We also present the Mellin matrix transform of the extended Wright hypergeometric matrix function. After that, we present some fractional calculus findings for these expanded Wright hypergeometric matrix functions. Lastly, we present several theorems of the extended Wright hypergeometric matrix function in fractional Kinetic equations.

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5

Álvarez-Nodarse, R., N. R. Quintero, and A. Ronveaux. "On the linearization problem involving Pochhammer symbols and their q-analogues." Journal of Computational and Applied Mathematics 107, no. 1 (1999): 133–46. http://dx.doi.org/10.1016/s0377-0427(99)00087-4.

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6

Srivastava, Rekha, Ritu Agarwal, and Sonal Jain. "A family of the incomplete hypergeometric functions and associated integral transform and fractional derivative formulas." Filomat 31, no. 1 (2017): 125–40. http://dx.doi.org/10.2298/fil1701125s.

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Recently, Srivastava et al. [Integral Transforms Spec. Funct. 23 (2012), 659-683] introduced the incomplete Pochhammer symbols that led to a natural generalization and decomposition of a class of hypergeometric and other related functions as well as to certain potentially useful closed-form representations of definite and improper integrals of various special functions of applied mathematics and mathematical physics. In the present paper, our aim is to establish several formulas involving integral transforms and fractional derivatives of this family of incomplete hypergeometric functions. As corollaries and consequences, many interesting results are shown to follow from our main results.

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7

Dr., Pranesh Kulkarni. "Analysis of Classical Special Beta & Gamma Functions in Engineering Mathematics and Physics." Indian Journal of Advanced Mathematics (IJAM) 5, no. 1 (2025): 35–37. https://doi.org/10.54105/ijam.A1195.05010425.

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<strong>Abstract: </strong>In many areas of applied mathematics, various types of Special functions have become essential tools for Scientists and engineers. Both Beta and Gamma functions are very important in calculus as complex integrals can be moderated into simpler form. In physics and engineering problems require a detailed knowledge of applied mathematics and an understanding of special functions such as gamma and beta functions. The topic of special functions is very important and it is constantly expanding with the existence of new problems in the applied Sciences in this article, we describe the basic theory of gamma and beta functions, their connections with each other and their applicability to engineering problems.to compute and depict scattering amplitude in Reggae trajectories. Our aim is to illustrate the extension of the classical beta function has many uses. It helps in providing new extensions of the beta distribution, providing new extensions of the Gauss hyper geometric functions and confluent hyper geometric function and generating relations, and extension of Riemann-Lowville derivatives. In this Article, we develop some elementary properties of the beta and gamma functions. We give more than one proof for some results. Often, one proof generalizes and others do not. We briefly discuss the finite field analogy of the gamma and beta functions. These are called Gauss and Jacobi sums and are important in number theory. We show how they can be used to prove Fermat's theorem that a prime of the form 4n + 1 is expressible as a sum of two squares. We also treat a simple multidimensional extension of a beta integral.

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8

Bedratyuk, L., and A. Bedratuyk. "The inverse and derivative connecting problems for some hypergeometric polynomials." Carpathian Mathematical Publications 10, no. 2 (2018): 235–47. http://dx.doi.org/10.15330/cmp.10.2.235-247.

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Given two polynomial sets $\{ P_n(x) \}_{n\geq 0},$ and $\{ Q_n(x) \}_{n\geq 0}$ such that $\deg ( P_n(x) ) = \deg ( Q_n(x) )=n.$ The so-called connection problem between them asks to find coefficients $\alpha_{n,k}$ in the expression $\displaystyle Q_n(x) =\sum_{k=0}^{n} \alpha_{n,k} P_k(x).$ The connection problem for different types of polynomials has a long history, and it is still of interest. The connection coefficients play an important role in many problems in pure and applied mathematics, especially in combinatorics, mathematical physics and quantum chemical applications. For the particular case $Q_n(x)=x^n$ the connection problem is called the inversion problem associated to $\{P_n(x)\}_{n\geq 0}.$ The particular case $Q_n(x)=P'_{n+1}(x)$ is called the derivative connecting problem for polynomial family $\{ P_n(x) \}_{n\geq 0}.$ In this paper, we give a closed-form expression of the inversion and the derivative coefficients for hypergeometric polynomials of the form $${}_2 F_1 \left[ \left. \begin{array}{c} -n, a \\ b \end{array} \right | z \right], {}_2 F_1 \left[ \left. \begin{array}{c} -n, n+a \\ b \end{array} \right | z \right], {}_2 F_1 \left[ \left. \begin{array}{c} -n, a \\ \pm n +b \end{array} \right | z \right],$$ where $\displaystyle {}_2 F_1 \left[ \left. \begin{array}{c} a, b \\ c \end{array} \right | z \right] =\sum_{k=0}^{\infty} \frac{(a)_k (b)_k}{(c)_k} \frac{z^k}{k!},$ is the Gauss hypergeometric function and $(x)_n$ denotes the Pochhammer symbol defined by $$\displaystyle (x)_n=\begin{cases}1, n=0, \\x(x+1)(x+2)\cdots (x+n-1) , n>0.\end{cases}$$ All polynomials are considered over the field of real numbers.

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9

Srivastava, Rekha. "Some Generalizations of Pochhammer’s Symbol and their Associated Families of Hypergeometric Functions and Hypergeometric Polynomials." Applied Mathematics & Information Sciences 7, no. 6 (2013): 2195–206. http://dx.doi.org/10.12785/amis/070609.

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10

Qureshi, Mohammad Idris, and Showkat Ahmad Dar. "Generalizations and applications of Srinivasa Ramanujan’s integral associated with infinite Fourier sine transforms in terms of Meijer’s G-function." Analysis 41, no. 3 (2021): 145–53. http://dx.doi.org/10.1515/anly-2018-0067.

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Abstract In this paper, we obtain analytical solutions of an unsolved integral 𝐑 S ⁢ ( m , n ) {\mathbf{R}_{S}(m,n)} of Srinivasa Ramanujan [S. Ramanujan, Some definite integrals connected with Gauss’s sums, Mess. Math. 44 1915, 75–86] with suitable convergence conditions in terms of Meijer’s G-function of one variable, by using Mellin–Barnes type contour integral representations of the sine function, Laplace transform method and some algebraic properties of Pochhammer’s symbol. Also, we have given some generalizations of Ramanujan’s integral 𝐑 S ⁢ ( m , n ) {\mathbf{R}_{S}(m,n)} in the form of integrals ℧ S * ⁢ ( υ , b , c , λ , y ) {\mho_{S}^{*}(\upsilon,b,c,\lambda,y)} , Ξ S ⁢ ( υ , b , c , λ , y ) {\Xi_{S}(\upsilon,b,c,\lambda,y)} , ∇ S ⁡ ( υ , b , c , λ , y ) {\nabla_{S}(\upsilon,b,c,\lambda,y)} and ℧ S ⁢ ( υ , b , λ , y ) {\mho_{S}(\upsilon,b,\lambda,y)} with suitable convergence conditions and solved them in terms of Meijer’s G-functions. Moreover, as applications of Ramanujan’s integral 𝐑 S ⁢ ( m , n ) {\mathbf{R}_{S}(m,n)} , the three new infinite summation formulas associated with Meijer’s G-function are obtained.

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11

Sahai, Vivek, and Ashish Verma. "On an extension of the generalized Pochhammer symbol and its applications to hypergeometric functions." Asian-European Journal of Mathematics 09, no. 03 (2016): 1650064. http://dx.doi.org/10.1142/s1793557116500649.

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The main object of this paper is to present a generalization of the Pochhammer symbol. We present some contiguous relations of this generalized Pochhammer symbol and use it to give an extension of the generalized hypergeometric function [Formula: see text]. Finally, we present some properties and generating functions of this extended generalized hypergeometric function.

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12

Choi, Junesang, Rakesh K. Parmar, and Purnima Chopra. "Extended Mittag-Leffler function and associated fractional calculus operators." Georgian Mathematical Journal 27, no. 2 (2020): 199–209. http://dx.doi.org/10.1515/gmj-2019-2030.

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AbstractMotivated mainly by certain interesting recent extensions of the generalized hypergeometric function [H. M. Srivastava, A. Çetinkaya and I. Onur Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 2014, 484–491] by means of the generalized Pochhammer symbol, we introduce here a new extension of the generalized Mittag-Leffler function. We then systematically investigate several properties of the extended Mittag-Leffler function including some basic properties, Mellin, Euler-Beta, Laplace and Whittaker transforms. Furthermore, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag-Leffler function are also investigated. Some interesting special cases of our main results are pointed out.

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13

Saboor, Abdus, Gauhar Rahman, Hazrat Ali, Kottakkaran Sooppy Nisar, and Thabet Abdeljawad. "Properties and Applications of a New Extended Gamma Function Involving Confluent Hypergeometric Function." Journal of Mathematics 2021 (February 24, 2021): 1–12. http://dx.doi.org/10.1155/2021/2491248.

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In this paper, a new confluent hypergeometric gamma function and an associated confluent hypergeometric Pochhammer symbol are introduced. We discuss some properties, for instance, their different integral representations, derivative formulas, and generating function relations. Different special cases are also considered.

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14

Masjed-Jamei, Mohammad, and Gradimir Milovanovic. "An extension of Pochhammer’s symbol and its application to hypergeometric functions." Filomat 31, no. 2 (2017): 207–15. http://dx.doi.org/10.2298/fil1702207m.

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By using a special property of the gamma function, we first define a productive form of gamma and beta functions and study some of their general properties in order to define a new extension of the Pochhammer symbol. We then apply this extended symbol for generalized hypergeometric series and study the convergence problem with some illustrative examples in this sense. Finally, we introduce two new extensions of Gauss and confluent hypergeometric series and obtain some of their general properties.

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15

Lin, Shy-Der, H. M. Srivastava, and Mu-Ming Wong. "Some applications of Srivastava’s theorem involving a certain family of generalized and extended hypergeometric polynomials." Filomat 29, no. 8 (2015): 1811–19. http://dx.doi.org/10.2298/fil1508811l.

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Recently, Srivastava et al. [H. M. Srivastava, M. A. Chaudhry and R. P. Agarwal, The incomplete Pochhammer symbols and their applications to hypergeometric and related functions, Integral Transforms Spec. Funct. 23 (2012), 659-683] introduced and initiated the study of many interesting fundamental properties and characteristics of a certain pair of potentially useful families of the so-called generalized incomplete hypergeometric functions. Ever since then there have appeared many closely-related works dealing essentially with notable developments involving various classes of generalized hypergeometric functions and generalized hypergeometric polynomials, which are defined by means of the corresponding incomplete and other novel extensions of the familiar Pochhammer symbol. Here, in this sequel to some of these earlier works, we derive several general families of hypergeometric generating functions by applying Srivastava?s Theorem. We also indicate various (known or new) special cases and consequences of the results presented in this paper.

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16

Burić, Tomislav. "On the Asymptotic Expansions of the (p,k)-Analogues of the Gamma Function and Associated Functions." Axioms 14, no. 1 (2025): 55. https://doi.org/10.3390/axioms14010055.

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General asymptotic expansion of the (p,k)-gamma function is obtained and various approaches to this expansion are studied. The numerical precision of the derived asymptotic formulas is shown and compared. Results are applied to the analogues of digamma and polygamma functions, and asymptotic expansion of the quotient of two (p,k)-gamma functions is also derived and analyzed. Various examples and application to the k-Pochhammer symbol are presented.

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17

Khan, Waseem Ahmad, Hassen Aydi, Musharraf Ali, Mohd Ghayasuddin, and Jihad Younis. "Construction of Generalized k-Bessel–Maitland Function with Its Certain Properties." Journal of Mathematics 2021 (November 20, 2021): 1–14. http://dx.doi.org/10.1155/2021/5386644.

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The main motive of this study is to present a new class of a generalized k -Bessel–Maitland function by utilizing the k -gamma function and Pochhammer k -symbol. By this approach, we deduce a few analytical properties as usual differentiations and integral transforms (likewise, Laplace transform, Whittaker transform, beta transform, and so forth) for our presented k -Bessel–Maitland function. Also, the k -fractional integration and k -fractional differentiation of abovementioned k -Bessel–Maitland functions are also pointed out systematically.

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18

Pal, Ankit, R. K. Jana, and A. K. Shukla. "Generalized Integral Transform and Fractional Calculus Involving Extended <sub>p</sub>R<sub>q</sub>(α β Ζ) Function." Journal of the Indian Mathematical Society 89, no. 1-2 (2022): 100. http://dx.doi.org/10.18311/jims/2022/29310.

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In this paper, we address an extended version of <em><sub>p</sub>R<sub>q</sub>(? ? ?)</em> function using k-Pochhammer symbol and study their classical properties and generalized integral transform. Further, we study Pathway fractional hypergeometric integral and fractional derivatives of the extended <em><sub>p</sub>R<sub>q</sub>(? ? ?)</em> function. Some special cases have also been illustrated.

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19

Rahman, Gauhar, Muhammad Samraiz, Manar A. Alqudah, and Thabet Abdeljawad. "Multivariate Mittag-Leffler function and related fractional integral operators." AIMS Mathematics 8, no. 6 (2023): 13276–93. http://dx.doi.org/10.3934/math.2023671.

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<abstract><p>In this paper, we describe a new generalization of the multivariate Mittag-Leffler (M-L) function in terms of generalized Pochhammer symbol and study its properties. We provide a few differential and fractional integral formulas for the generalized multivariate M-L function. Furthermore, by using the generalized multivariate M-L function in the kernel, we present a new generalization of the fractional integral operator. Finally, we describe some fundamental characteristics of generalized fractional integrals.</p></abstract>

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20

González-Santander, Juan Luis, and Fernando Sánchez Lasheras. "Finite and Infinite Hypergeometric Sums Involving the Digamma Function." Mathematics 10, no. 16 (2022): 2990. http://dx.doi.org/10.3390/math10162990.

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We calculate some finite and infinite sums containing the digamma function in closed form. For this purpose, we differentiate selected reduction formulas of the hypergeometric function with respect to the parameters applying some derivative formulas of the Pochhammer symbol. Additionally, we compare two different differentiation formulas of the generalized hypergeometric function with respect to the parameters. For some particular cases, we recover some results found in the literature. Finally, all the results have been numerically checked.

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21

Bakhet, Ahmed, Abd-Allah Hyder, Areej A. Almoneef, Mohamed Niyaz, and Ahmed H. Soliman. "On New Matrix Version Extension of the Incomplete Wright Hypergeometric Functions and Their Fractional Calculus." Mathematics 10, no. 22 (2022): 4371. http://dx.doi.org/10.3390/math10224371.

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Through this article, we will discuss a new extension of the incomplete Wright hypergeometric matrix function by using the extended incomplete Pochhammer matrix symbol. First, we give a generalization of the extended incomplete Wright hypergeometric matrix function and state some integral equations and differential formulas about it. Next, we obtain some results about fractional calculus of these extended incomplete Wright hypergeometric matrix functions. Finally, we discuss an application of the extended incomplete Wright hypergeometric matrix function in the kinetic equations.

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22

Awad, Mohamed M. "On a Generalization of Kummer’s Second-Type F 1 1 and F 2 2." Mathematical Problems in Engineering 2021 (June 5, 2021): 1–11. http://dx.doi.org/10.1155/2021/5531388.

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The aim of this paper is to establish a general form of Kummer’s second-type summation theory. By defining a new form for the divided of the Pochhammer symbol d + i n / d n , we can develop a general form of Kummer’s second-type summation theorem as e − x / 2 F 2 2 a , i + d ; x b + l , d ; in the form of a sum of e − x / 2 F 1 1 a ; x b + i ; for i , l = 0,1,2 , … , Then, some properties of the generalized Kummer’s second-type summation theorem can yield a number of known and novel results.

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23

GEZER, Halil, and Cem KAANOGLU. "On the extended Wright hypergeometric matrix function and its properties." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 72, no. 3 (2023): 606–17. http://dx.doi.org/10.31801/cfsuasmas.1147745.

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Recently, Bakhet et al. [9] presented the Wright hypergeometric matrix function $_{2}R_{1}^{(\tau )}(A,B;C;z)$ and derived several properties. Abdalla [6] has since applied fractional operators to this function. In this paper, with the help of the generalized Pochhammer matrix symbol $(A;B)_{n}$ and the generalized beta matrix function $\mathcal{B}(P,Q;\mathbb{X})$, we introduce and study an extended form of the Wright hypergeometric matrix function, $_{2}R_{1}^{(\tau )}((A,\mathbb{A}),B;C;z;\mathbb{X}).$ We establish several potentially useful results for this extended form, such as integral representations and fractional derivatives. We also derive some properties of the corresponding incomplete extended Wright hypergeometric matrix function.

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24

Melas, A., and N. Bagis. "A Pochhammer type sampling reconstruction of analytic functions." Journal of Mathematical Analysis and Applications 373, no. 1 (2011): 214–26. http://dx.doi.org/10.1016/j.jmaa.2010.06.040.

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25

Zayed, E. M. E., and K. A. E. Alurrfi. "The Generalized Projective Riccati Equations Method for Solving Nonlinear Evolution Equations in Mathematical Physics." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/259190.

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We apply the generalized projective Riccati equations method to find the exact traveling wave solutions of some nonlinear evolution equations with any-order nonlinear terms, namely, the nonlinear Pochhammer-Chree equation, the nonlinear Burgers equation and the generalized, nonlinear Zakharov-Kuznetsov equation. This method presents wider applicability for handling many other nonlinear evolution equations in mathematical physics.

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26

Kukla, Stanisław, and Urszula Siedlecka. "On Solutions of the Initial Value Problem for the Three-Term Fractional Differential Equation with Caputo Derivatives." Symmetry 12, no. 8 (2020): 1355. http://dx.doi.org/10.3390/sym12081355.

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In this paper, two forms of an exact solution and an analytical–numerical solution of the three-term fractional differential equation with the Caputo derivatives are presented. The Prabhakar function and an asymptotic expansion are utilized to present the double series solution. Using properties of the Pochhammer symbol, a solution is obtained in the form of an infinite series of generalized hypergeometric functions. As an alternative for the series solutions of the fractional commensurate equation, a solution received by an analytical–numerical method based on the Laplace transform technique is proposed. This solution is obtained in the form of a finite sum of the Mittag-Leffler type functions. Numerical examples and a discussion are presented.

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27

LI, JIBIN, and GUANRONG CHEN. "EXACT TRAVELING WAVE SOLUTIONS AND THEIR BIFURCATIONS FOR THE GENERALIZED POCHHAMMER–CHREE EQUATIONS." International Journal of Bifurcation and Chaos 22, no. 09 (2012): 1250233. http://dx.doi.org/10.1142/s0218127412502331.

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By using the method of dynamical systems to study the generalized Pochhammer–Chree equations, the dynamics of traveling wave solutions are characterized under different parameter conditions. Some exact parametric representations of the traveling wave solutions are obtained. Thus, many results reported in the literature can be completed.

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28

Lee, C. K. B., R. C. Crawford, K. A. Mann, P. Coleman, and C. Petersen. "Evidence of higher Pochhammer-Chree modes in an unsplit Hopkinson bar." Measurement Science and Technology 6, no. 7 (1995): 853–59. http://dx.doi.org/10.1088/0957-0233/6/7/001.

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29

Zuo, Jin-Ming. "Application of the extended -expansion method to solve the Pochhammer–Chree equations." Applied Mathematics and Computation 217, no. 1 (2010): 376–83. http://dx.doi.org/10.1016/j.amc.2010.05.072.

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30

HERCZYNSKI, ANDRZEJ, and ROBERT T. FOLK. "ORTHOGONALITY CONDITION FOR THE POCHHAMMER-CHREE MODES." Quarterly Journal of Mechanics and Applied Mathematics 42, no. 4 (1989): 523–36. http://dx.doi.org/10.1093/qjmam/42.4.523.

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31

Shapiro, Rob. "Strange Symbol." New England Review 42, no. 4 (2021): 64–65. http://dx.doi.org/10.1353/ner.2021.0103.

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32

Abdalla, Mohamed, and Muajebah Hidan. "Investigation of the k-Analogue of Gauss Hypergeometric Functions Constructed by the Hadamard Product." Symmetry 13, no. 4 (2021): 714. http://dx.doi.org/10.3390/sym13040714.

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Traditionally, the special function theory has many applications in various areas of mathematical physics, economics, statistics, engineering, and many other branches of science. Inspired by certain recent extensions of the k-analogue of gamma, the Pochhammer symbol, and hypergeometric functions, this work is devoted to the study of the k-analogue of Gauss hypergeometric functions by the Hadamard product. We give a definition of the Hadamard product of k-Gauss hypergeometric functions (HPkGHF) associated with the fourth numerator and two denominator parameters. In addition, convergence properties are derived from this function. We also discuss interesting properties such as derivative formulae, integral representations, and integral transforms including beta transform and Laplace transform. Furthermore, we investigate some contiguous function relations and differential equations connecting the HPkGHF. The current results are more general than previous ones. Moreover, the proposed results are useful in the theory of k-special functions where the hypergeometric function naturally occurs.

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33

Zhao, Yan, and Weiguo Zhang. "Qualitative Analysis and Travelling Wave Solutions for the Generalized Pochhammer-Chree Equation with a Dissipation Term." Studies in Applied Mathematics 121, no. 4 (2008): 369–94. http://dx.doi.org/10.1111/j.1467-9590.2008.00420.x.

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34

Rust, Richard Dilworth. "Light: A Masterful Symbol." Journal of the Book of Mormon and Other Restoration Scripture 20, no. 1 (2011): 52–65. http://dx.doi.org/10.5406/jbookmormotheres.20.1.0052.

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35

Plonka, Gerlind. "Two-scale symbol and autocorrelation symbol for B-splines with multiple knots." Advances in Computational Mathematics 3, no. 1-2 (1995): 1–22. http://dx.doi.org/10.1007/bf02431993.

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36

Gerbracht, Eberhard H. A. "Symbol-crunching the Harborth graph." Advances in Applied Mathematics 47, no. 2 (2011): 276–81. http://dx.doi.org/10.1016/j.aam.2010.09.003.

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37

Wazwaz, Abdul-Majid. "The tanh–coth and the sine–cosine methods for kinks, solitons, and periodic solutions for the Pochhammer–Chree equations." Applied Mathematics and Computation 195, no. 1 (2008): 24–33. http://dx.doi.org/10.1016/j.amc.2007.04.066.

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38

Ellis, Keaton, Baian Liu, and Attila Sali. "Multi-symbol forbidden configurations." Discrete Applied Mathematics 276 (April 2020): 24–36. http://dx.doi.org/10.1016/j.dam.2019.11.010.

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39

Battail, Gérard. "Symbol-by-symbol Decoding as an Approximation of Word-by-word Decoding." Electronic Notes in Discrete Mathematics 6 (April 2001): 195–210. http://dx.doi.org/10.1016/s1571-0653(04)00171-4.

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40

Mirotin, Adolf R. "On the description of multidimensional normal Hausdorff operators on Lebesgue spaces." Forum Mathematicum 32, no. 1 (2020): 111–19. http://dx.doi.org/10.1515/forum-2019-0097.

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AbstractHausdorff operators originated from some classical summation methods. Now this is an active research field. In the present article, a spectral representation for multidimensional normal Hausdorff operator is given. We show that normal Hausdorff operator in {L^{2}(\mathbb{R}^{n})} is unitary equivalent to the operator of multiplication by some matrix-valued function (its matrix symbol) in the space {L^{2}(\mathbb{R}^{n};\mathbb{C}^{2^{n}})}. Several corollaries that show that properties of a Hausdorff operator are closely related to the properties of its symbol are considered. In particular, the norm and the spectrum of such operators are described in terms of the symbol.

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41

Kastis, Eleftherios, Derek Kitson, and John E. McCarthy. "Symbol functions for symmetric frameworks." Journal of Mathematical Analysis and Applications 497, no. 2 (2021): 124895. http://dx.doi.org/10.1016/j.jmaa.2020.124895.

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Erfiani, Ni Made Diana, Nyoman Tri Sukarsih, and I. Gusti Agung Sri Rwa Jayantini. "Dancing Cross." Mudra Jurnal Seni Budaya 38, no. 4 (2023): 395–406. http://dx.doi.org/10.31091/mudra.v38i4.2439.

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Religious symbol is the generic word for all kinds of religious signs of the divine. It includes myths and religious narratives, theological ideas, particular notions such as karma or sin which are defined in one or more symbol systems, religious acts such as liturgies or private meditations that move through and overlay various symbol systems, as well as architectural and artistic symbols with religious content, books, songs, devotional objects, and the like. Generally, it can be referred to a religious object and can bear a religious meaning. This study aims to identify the meaning of a contextualized religious symbol in the Bali Church which is known as ‘the dancing cross’. The theory applied in this study is semiotic theory which in this case is structural semiotics. The source of the data are texts that discuss about the cross symbol as well as the results of interviews with the creator of the dancing cross symbol, Nyoman Darsane. This study confirms that the dancing cross, which is widely used by Christians who are members of the Protestant Christian Church in Bali is considered as the product of regionalization or contextualization of the cross which is the crux of Christian life and worship. The shape of the dancing cross is transformed from the universal shape of the cross which is consisted of two bars leading to a T-shaped cross which base stem is longer than the other three arms but crooked at the bottom like a dancer who is in left agem position. The results of the analysis through the utilization of structural semiotics in terms of syntagmatic and paradigmatic relations revealed some important points namely the contrastive meaning of the cross before and after the time of Jesus Christ as the sign of shame which covers a curse, a stumbling block and foolishness and the sign to be glorified which covers salvation victory and obedience. Other important points to be revealed are the contextual shape of the cross that can be aligned as a dancer who is in left agem position and the contextual meaning of the cross which leads to the sign of obedience of God’s people or the disciples to carry or dance the cross in accordance with the example given by Jesus Christ.

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Sun, Zhi-Hong. "Consecutive numbers with the same Legendre symbol." Proceedings of the American Mathematical Society 130, no. 9 (2002): 2503–7. http://dx.doi.org/10.1090/s0002-9939-02-06600-5.

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Qing, Lin. "Imbedded operators with finite Blaschke product symbol." Acta Mathematica Sinica 6, no. 1 (1990): 72–79. http://dx.doi.org/10.1007/bf02108866.

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Gu, Caixing, Erin Rizzie, and Jonathan Shapiro. "Adjoints of composition operators with irrational symbol." Proceedings of the American Mathematical Society 148, no. 1 (2019): 145–55. http://dx.doi.org/10.1090/proc/13340.

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Hartl, Richard F., Andreas J. Novak, Ambar G. Rao, and Suresh P. Sethi. "Dynamic pricing of a status symbol." Nonlinear Analysis: Theory, Methods & Applications 63, no. 5-7 (2005): e2301-e2314. http://dx.doi.org/10.1016/j.na.2005.03.025.

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Porter, Curtis, and Igor Zelenko. "Absolute parallelism for 2-nondegenerate CR structures via bigraded Tanaka prolongation." Journal für die reine und angewandte Mathematik (Crelles Journal) 2021, no. 777 (2021): 195–250. http://dx.doi.org/10.1515/crelle-2021-0012.

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Abstract This article is devoted to the local geometry of everywhere 2-nondegenerate CR manifolds M of hypersurface type. An absolute parallelism for such structures was recently constructed independently by Isaev and Zaitsev, Medori and Spiro, and Pocchiola in the minimal possible dimension ( dim ⁡ M = 5 {\dim M=5} ), and for dim ⁡ M = 7 {\dim M=7} in certain cases by the first author. In the present paper, we develop a bigraded (i.e., ℤ × ℤ {\mathbb{Z}\times\mathbb{Z}} -graded) analog of Tanaka’s prolongation procedure to construct an absolute parallelism for these CR structures in arbitrary (odd) dimension with Levi kernel of arbitrary admissible dimension. We introduce the notion of a bigraded Tanaka symbol – a complex bigraded vector space – containing all essential information about the CR structure. Under the additional regularity assumption that the symbol is a Lie algebra, we define a bigraded analog of the Tanaka universal algebraic prolongation, endowed with an anti-linear involution, and prove that for any CR structure with a given regular symbol there exists a canonical absolute parallelism on a bundle whose dimension is that of the bigraded universal algebraic prolongation. Moreover, we show that for each regular symbol there is a unique (up to local equivalence) such CR structure whose algebra of infinitesimal symmetries has maximal possible dimension, and the latter algebra is isomorphic to the real part of the bigraded universal algebraic prolongation of the symbol. In the case of 1-dimensional Levi kernel we classify all regular symbols and calculate their bigraded universal algebraic prolongations. In this case, the regular symbols can be subdivided into nilpotent, strongly non-nilpotent, and weakly non-nilpotent. The bigraded universal algebraic prolongation of strongly non-nilpotent regular symbols is isomorphic to the complex orthogonal algebra 𝔰 ⁢ 𝔬 ⁢ ( m , ℂ ) {\mathfrak{so}(m,\mathbb{C})} , where m = 1 2 ⁢ ( dim ⁡ M + 5 ) {m=\tfrac{1}{2}(\dim M+5)} . Any real form of this algebra – except 𝔰 ⁢ 𝔬 ⁢ ( m ) {\mathfrak{so}(m)} and 𝔰 ⁢ 𝔬 ⁢ ( m - 1 , 1 ) {\mathfrak{so}(m-1,1)} – corresponds to the real part of the bigraded universal algebraic prolongation of exactly one strongly non-nilpotent regular CR symbol. However, for a fixed dim ⁡ M ≥ 7 {\dim M\geq 7} the dimension of the bigraded universal algebraic prolongations of all possible regular CR symbols achieves its maximum on one of the nilpotent regular symbols, and this maximal dimension is 1 4 ⁢ ( dim ⁡ M - 1 ) 2 + 7 {\frac{1}{4}(\dim M-1)^{2}+7} .

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Hammond, Christopher, Jennifer Moorhouse, and Marian E. Robbins. "Adjoints of composition operators with rational symbol." Journal of Mathematical Analysis and Applications 341, no. 1 (2008): 626–39. http://dx.doi.org/10.1016/j.jmaa.2007.10.039.

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Gilbert, John E., and Andrea R. Nahmod. "Bilinear operators with non-smooth symbol, I." Journal of Fourier Analysis and Applications 7, no. 5 (2001): 435–67. http://dx.doi.org/10.1007/bf02511220.

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Yuli Astuti, Amelia, and Suci Rahmadani. "Linguistic Signs of The Symbolic Meaning in Tabuik Pariaman West Sumatra." Jurnal Ilmiah Pendidikan Scholastic 7, no. 3 (2023): 74–81. http://dx.doi.org/10.36057/jips.v7i3.639.

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This research is entitled linguistic on the meaning of tabuik symboik in Pariaman, West Sumatra. It explains the sign of the linguistic meaning of pen and Omen, denotative and connotative meaning on Tabuic symbols. It aims to improve the order of pens and signs and enhance the denotative and connotative meanings in Tabuic symbols. Based on these sources, the data were collected by observation method with interview and photo techniques. The analysis is carried out by semiotics, signs of linguistic meaning and the sequence of presentation methods. The Data were analyzed by associating them with concepts discovered by Chandler (2007).Saussure's theory was to classify the pen and the mark.Chandler's theory is to classify denotative and connotative meanings. The analysis results are presented descriptively. After analyzing the data, it was found that there are several specifications of the meaning of the Tabuik symbol. This research shows that the Tabuik symbol has a high cultural and religious meaning and value, especially in Pariaman . Tabuik has the form of a sign and a sign and the meaning we can see from the form and symbol of Tabuik.

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