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Journal articles on the topic 'Hypergeometric Kummer's function'

Author: Grafiati

Published: 4 June 2025

Last updated: 23 June 2025

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1

Qureshi, M. I., M. Sadiq Khan, M. A. Pathan, and N. U. Khan. "Some multivariable Gaussian hypergeometric extensions of the Preece theorem." ANZIAM Journal 48, no. 1 (2006): 143–50. http://dx.doi.org/10.1017/s1446181100003473.

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AbstractSome generalisations of the Preece theorem involving the product of two Kummer's functions 1F1 are obtained using Dixon's theorem and some well-known identities. Its special cases yield various new transformations and reduction formulae involving Pathan's quadruple hypergeometric function and Srivastava's quadruple hypergeometric function F(4) and triple hypergeometric function F(3). Some known results of Preece, Pathan and Bailey are also obtained as special cases.

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2

Milovanovic, Gradimir, Rakesh Parmar, and Arjun Rathie. "A study of generalized summation theorems for the series 2F1 with an applications to Laplace transforms of convolution type integrals involving Kummer's functions 1F1." Applicable Analysis and Discrete Mathematics 12, no. 1 (2018): 257–72. http://dx.doi.org/10.2298/aadm171017002m.

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Motivated by recent generalizations of classical theorems for the series 2F1 [Integral Transform. Spec. Funct. 229(11), (2011), 823-840] and interesting Laplace transforms of Kummer's confluent hypergeometric functions obtained by Kim et al. [Math. Comput. Modelling 55 (2012), 1068-1071], first we express generalized summations theorems in explicit forms and then by employing these, we derive various new and useful Laplace transforms of convolution type integrals by using product theorem of the Laplace transforms for a pair of Kummer's confluent hypergeometric function.

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3

Mathews, W. N., M. A. Esrick, Z. Y. Teoh, and J. K. Freericks. "A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions." Condensed Matter Physics 25, no. 3 (2022): 33203. http://dx.doi.org/10.5488/cmp.25.33203.

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The confluent hypergeometric equation, also known as Kummer's equation, is one of the most important differential equations in physics, chemistry, and engineering. Its two power series solutions are the Kummer function, M(a,b,z), often referred to as the confluent hypergeometric function of the first kind, and M ≡ z1-bM(1+a-b, 2-b,z), where a and b are parameters that appear in the differential equation. A third function, the Tricomi function, U(a,b,z), sometimes referred to as the confluent hypergeometric function of the second kind, is also a solution of the confluent hypergeometric equation that is routinely used. Contrary to common procedure, all three of these functions (and more) must be considered in a search for the two linearly independent solutions of the confluent hypergeometric equation. There are situations, when a, b, and a - b are integers, where one of these functions is not defined, or two of the functions are not linearly independent, or one of the linearly independent solutions of the differential equation is different from these three functions. Many of these special cases correspond precisely to cases needed to solve problems in physics. This leads to significant confusion about how to work with confluent hypergeometric equations, in spite of authoritative references such as the NIST Digital Library of Mathematical Functions. Here, we carefully describe all of the different cases one has to consider and what the explicit formulas are for the two linearly independent solutions of the confluent hypergeometric equation. The procedure to properly solve the confluent hypergeometric equation is summarized in a convenient table. As an example, we use these solutions to study the bound states of the hydrogenic atom, correcting the standard treatment in textbooks. We also briefly consider the cutoff Coulomb potential. We hope that this guide will aid physicists to properly solve problems that involve the confluent hypergeometric differential equation.

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4

Morita, Tohru. "Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Nonstandard Analysis, in Terms of the Green's Function." Journal of Advances in Mathematics and Computer Science 39, no. 1 (2024): 20–28. http://dx.doi.org/10.9734/jamcs/2024/v39i11859.

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Discussions are presented by Morita and Sato on the problem of obtaining the particular solution of an inhomogeneous differential equation with polynomial coefficients in terms of the Green's function. In a preceding paper, solution is given without using the Green's function, on the basis of nonstandard analysis, for a restricted class of inhomogeneous terms. In the present paper, the corresponding solutions are given in terms of the Green's function. It is applied to Kummer's and the hypergeometric differential equation.

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5

Mohammed, Asmaa Orabi, Medhat A. Rakha, Mohammed M. Awad, and Arjun K. Rathie. "On several new Laplace transforms of generalized hypergeometric functions 2F2(x)." Boletim da Sociedade Paranaense de Matemática 39, no. 4 (2021): 97–109. http://dx.doi.org/10.5269/bspm.42207.

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By employing generalizations of Gauss's second, Bailey's and Kummer's summation theorems obtained earlier by Rakha and Rathie, we aim to establish unknown Laplace transform of six rather general formulas of generalized hypergeometric function 2F2[a,b;c,d;x]. The results obtained in this paper are simple, interesting, easily established and may be useful in theoretical physics, engineering and mathematics. Results obtained earlier by Kim et al. and Choi and Rathie follow special cases of our main findings.

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6

Moothathu, T. S. K. "The Best Estimators of Quantiles and the Three Means of the Pareto Distribution." Calcutta Statistical Association Bulletin 35, no. 3-4 (1986): 111–22. http://dx.doi.org/10.1177/0008068319860301.

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In this paper we develop the uniformly minimum variance unbiased (best) estimators of the quantiles, mean, geometric mean and harmonic mean of the Pareto distribuion in the case when both the shape parameter a and the scale parameter k are unknown and in cases when one of them alone is unknown. The best estimates are in terms of the Bessel Function o F1 and Kummer's function 1 F1. The variance of the best estimator are found out, which are in terms of F2 , the Appell function of second kind and ψ2 , a confluent hypergeometric function of two variables. Further we prove that every best estimator of this paper is strongly consistent.

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7

Guillemin, Fabrice, and Didier Pinchon. "Continued Fraction Analysis of the Duration of an Excursion in an M/M/∞ System." Journal of Applied Probability 35, no. 1 (1998): 165–83. http://dx.doi.org/10.1239/jap/1032192560.

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We show in this paper how the Laplace transform θ* of the duration θ of an excursion by the occupation process {Λt} of an M/M/∞ system above a given threshold can be obtained by means of continued fraction analysis. The representation of θ* by a continued fraction is established and the [m−1/m] Padé approximants are computed by means of well known orthogonal polynomials, namely associated Charlier polynomials. It turns out that the continued fraction considered is an S fraction and as a consequence the Stieltjes transform of some spectral measure. Then, using classic asymptotic expansion properties of hypergeometric functions, the representation of the Laplace transform θ* by means of Kummer's function is obtained. This allows us to recover an earlier result obtained via complex analysis and the use of the strong Markov property satisfied by the occupation process {Λt}. The continued fraction representation enables us to further characterize the distribution of the random variable θ.

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8

Guillemin, Fabrice, and Didier Pinchon. "Continued Fraction Analysis of the Duration of an Excursion in an M/M/∞ System." Journal of Applied Probability 35, no. 01 (1998): 165–83. http://dx.doi.org/10.1017/s0021900200014765.

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We show in this paper how the Laplace transform θ* of the duration θ of an excursion by the occupation process {Λ t } of an M/M/∞ system above a given threshold can be obtained by means of continued fraction analysis. The representation of θ* by a continued fraction is established and the [m−1/m] Padé approximants are computed by means of well known orthogonal polynomials, namely associated Charlier polynomials. It turns out that the continued fraction considered is an S fraction and as a consequence the Stieltjes transform of some spectral measure. Then, using classic asymptotic expansion properties of hypergeometric functions, the representation of the Laplace transform θ* by means of Kummer's function is obtained. This allows us to recover an earlier result obtained via complex analysis and the use of the strong Markov property satisfied by the occupation process {Λ t }. The continued fraction representation enables us to further characterize the distribution of the random variable θ.

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9

Poudel, Madhav Prasad, Harsh Vardhan Harsh, Narayan Prasad Pahari, and Dinesh Panthi. "Kummer’s Theorems, Popular Solutions and Connecting Formulas on Hypergeometric Function." Journal of Nepal Mathematical Society 6, no. 1 (2023): 48–56. http://dx.doi.org/10.3126/jnms.v6i1.57413.

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The hypergeometric series is an extension of the geometric series. The confluent hypergeometric function is the solution of the hypergeometric differential equation [θ(θ +b−1)−z(θ +a)]w = 0. Kummer’s first formula and Kummer’s second formula are of significant importance in solving the hypergeometric differential equations. Kummer has developed six solutions for the differential equation and twenty connecting formulas during the period of 1865-1866. Each connecting formula consist of a solution expressed as the combination of two other solutions. Recently in 2021, these solutions were extensively used by Schweizer [13] in practical problems specially in Physics. Here we extend the connecting formulas obtained by Kummer to obtain the other six solutions w1(z), w2(z), w3(z), w4(z), w5(z) and w6(z) as the combination of three solutions.

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10

Yuriy, Khomyak, Naumenko Ievgeniia, Zheglova Victoriia, and Popov Vadim. "MINIMIZING THE MASS OF A FLAT BOTTOM OF CYLINDRICAL APPARATUS." Eastern-European Journal of Enterprise Technologies 2, no. 1 (92) (2018): 42–50. https://doi.org/10.15587/1729-4061.2018.126141.

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In the bodies of cylindrical apparatuses that operate under pressure, one of the weak elements is a flat bottom whose thickness is increased by 4…5 times in comparison with the wall thickness. This is due to the fact that the bottom is exposed to a more unfavorable bending deformation compared to the wall that «works» on stretching. In order to reduce specific metal consumption for the bottom, we propose the optimization of the shape of a radial cross-section by a rational redistribution of the material: to increase thickness of the bottom in the region of its contact with the wall and to significantly reduce it in the central zone. To describe a variable thickness of the bottom, we applied the Gauss equation with an arbitrary parameter that determines the intensity of change in the thickness in radial direction. We have obtained a general solution to the differential equation of the problem on bending a bottom at a given law of change in its thickness, which is represented using the hypergeometric Kummer’s functions. A technique for concretizing the resulting solution was proposed and implemented, based on the application of conditions of contact between a cylindrical shell and a bottom. The solution derived was used to minimize the mass of the bottom. We have designed a zone of transition from the bottom to the wall whose strength was verified by the method of finite elements under actual conditions

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11

Chandel, R. C. Singh, and Hemant Kumar. "CONTOUR INTEGRAL REPRESENTATIONS OF TWO VARIABLE GENERALIZED HYPERGEOMETRIC FUNCTION OF SRIVASTAVA AND DAOUST WITH THEIR APPLICATIONS TO INITIAL VALUE PROBLEMS OF ARBITRARY ORDER." Jnanabha 50, no. 01 (2020): 232–42. http://dx.doi.org/10.58250/jnanabha.2020.50122.

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In this paper, we establish two contour integral representations involving Mittag - Leffler functions (i) for a two variable generalized hypergeometric function of Srivastava and Daoust and (ii) a sum of the Kummer’s confluent hypergeometric functions. Then, we make their appeal to obtain the contour integrals for many generating functions and bilateral generating relations. Further, in development and extensions of fractional calculus, we obtain various relations of contour integrals with fractional derivatives and integral operators to use them in solving of any order initial value problems

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12

Yan, Zhimin. "A Class of Generalized Hypergeometric Functions in Several Variables." Canadian Journal of Mathematics 44, no. 6 (1992): 1317–38. http://dx.doi.org/10.4153/cjm-1992-079-x.

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AbstractWe study a class of generalized hypergeometric functions in several variables introduced by A. Korânyi. It is shown that the generalized Gaussian hypergeometric function is the unique solution of a system partial differential equations. Analogues of some classical results such as Kummer relations and Euler integral representations are established. Asymptotic behavior of generalized hypergeometric functions is obtained which includes some known estimates.

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13

Oros, Georgia Irina, Gheorghe Oros, and Ancuța Maria Rus. "Applications of Confluent Hypergeometric Function in Strong Superordination Theory." Axioms 11, no. 5 (2022): 209. http://dx.doi.org/10.3390/axioms11050209.

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In the research presented in this paper, confluent hypergeometric function is embedded in the theory of strong differential superordinations. In order to proceed with the study, the form of the confluent hypergeometric function is adapted taking into consideration certain classes of analytic functions depending on an extra parameter previously introduced related to the theory of strong differential subordination and superordination. Operators previously defined using confluent hypergeometric function, namely Kummer–Bernardi and Kummer–Libera integral operators, are also adapted to those classes and strong differential superordinations are obtained for which they are the best subordinants. Similar results are obtained regarding the derivatives of the operators. The examples presented at the end of the study are proof of the applicability of the original results.

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14

Qureshi, Mohd Idris, Junesang Choi, and Tafaz Rahman Shah. "Certain Generalizations of Quadratic Transformations of Hypergeometric and Generalized Hypergeometric Functions." Symmetry 14, no. 5 (2022): 1073. http://dx.doi.org/10.3390/sym14051073.

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There have been numerous investigations on the hypergeometric series 2F1 and the generalized hypergeometric series pFq such as differential equations, integral representations, analytic continuations, asymptotic expansions, reduction cases, extensions of one and several variables, continued fractions, Riemann’s equation, group of the hypergeometric equation, summation, and transformation formulae. Among the various approaches to these functions, the transformation formulae for the hypergeometric series 2F1 and the generalized hypergeometric series pFq are significant, both in terms of applications and theory. The purpose of this paper is to establish a number of transformation formulae for pFq, whose particular cases would include Gauss’s and Kummer’s quadratic transformation formulae for 2F1, as well as their two extensions for 3F2, by making advantageous use of a recently introduced sequence and some techniques commonly used in dealing with pFq theory. The pFq function, which is the most significant function investigated in this study, exhibits natural symmetry.

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15

Srivastava, Hari M. "Some Double Integrals Stemming from the Boltzmann Equation in the Kinetic Theory of Gasses." European Journal of Pure and Applied Mathematics 15, no. 3 (2022): 810–20. http://dx.doi.org/10.29020/nybg.ejpam.v15i3.4429.

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The main object of this article is to revisit a certain double integral involving Kummer’s confluent hypergeometric function 1F1 , which arose in the study of the collision terms of the celebrated Boltzmann equation in the kinetic theory of gases. Here, in this article, we propose to investigate some novel extensions and generalizations of this family of double integrals. We also point out some relevant connections of the results, which are presented here, with other related recent developments in the theory and applications of hypergeometric functions.

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16

Poudel, Madhav Prasad, Narayan Prasad Pahari, Ganesh Basnet, and Resham Poudel. "Connection Formulas on Kummer’s Solutions and their Extension on Hypergeometric Function." Nepal Journal of Mathematical Sciences 4, no. 2 (2023): 83–88. https://doi.org/10.3126/njmathsci.v4i2.60177.

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Hypergeometric functions are transcendental functions that are applicable in various branches of mathematics, physics, and engineering. They are solutions to a class of differential equations called hypergeometric differential equations. Kummer obtained six solutions for the hypergeometric differential equation and twenty connection formulae. This research work has extended those connection formulas to other six solutions y1(x), y2(x), y3(x), y4(x), y5(x) and y6(x) show that each solution can be expressed in terms of linear relationship among three of the other solutions.

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17

Oros, Georgia Irina. "New Conditions for Univalence of Confluent Hypergeometric Function." Symmetry 13, no. 1 (2021): 82. http://dx.doi.org/10.3390/sym13010082.

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Since in many particular cases checking directly the conditions from the definitions of starlikeness or convexity of a function can be difficult, in this paper we use the theory of differential subordination and in particular the method of admissible functions in order to determine conditions of starlikeness and convexity for the confluent (Kummer) hypergeometric function of the first kind. Having in mind the results obtained by Miller and Mocanu in 1990 who used a,c∈R, for the confluent (Kummer) hypergeometric function, in this investigation a and c complex numbers are used and two criteria for univalence of the investigated function are stated. An example is also included in order to show the relevance of the original results of the paper.

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18

Khan, Nabiullah, Rakibul Sk, and Saddam Husain. "Certain Results on Extended Beta and Related Functions Using Matrix Arguments." Journal of New Theory, no. 49 (December 31, 2024): 16–29. https://doi.org/10.53570/jnt.1534850.

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In this study, we present and explore extended beta matrix functions (EBMFs) and their key properties. By utilizing the beta matrix function (BMF), we introduce novel extensions of the Gauss hypergeometric matrix function (GHMF) and Kummer hypergeometric matrix function (KHMF). We delve into their integral representations, recurrence relations, transformation properties, and differential formulas. Additionally, we investigate their statistical applications, mainly focusing on the beta distribution, and derive expressions for the mean, variance, and moment-generating functions. Furthermore, we apply EBMFs to develop the Appell matrix function (AMF) and Lauricella matrix function (LMF) and their integral forms.

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19

Miller, A. R., and R. B. Paris. "A Generalised Kummer-Type Transformation for the pFp(x) Hypergeometric Function." Canadian Mathematical Bulletin 55, no. 3 (2012): 571–78. http://dx.doi.org/10.4153/cmb-2011-095-6.

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AbstractIn a recent paper, Miller derived a Kummer-type transformation for the generalised hypergeometric function pFp(x) when pairs of parameters differ by unity, by means of a reduction formula for a certain Kampé de Fériet function. An alternative and simpler derivation of this transformation is obtained here by application of the well-known Kummer transformation for the confluent hypergeometric function corresponding to p = 1.

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20

Ernst, Thomas, and Per W. Karlsson. "Corollaries and multiple extensions of Gessel and Stanton hypergeometric summation formulas." Acta et Commentationes Universitatis Tartuensis de Mathematica 25, no. 1 (2021): 21–31. http://dx.doi.org/10.12697/acutm.2021.25.02.

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We find some new simple hypergeometric formulas in the footsteps of the important article by Gessel and Stanton. These are multiple reduction formulas, multiple summation formulas, as well as multiple transformation formulas for special Kampé de Fériet functions and Appell functions. The hypergeometric summation formulas have special function arguments in Q and parameter values in N or C. The proofs use Pfaff-Kummer transformation, Euler transformation, or an improved form of Slater reversion.

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21

El-Shahed, Moustafa, and Ahmed Salem. "An Extension of Wright Function and Its Properties." Journal of Mathematics 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/950728.

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The paper is devoted to the study of the functionWα,βγ,δ(z), which is an extension of the classical Wright function and Kummer confluent hypergeometric function. The properties ofWα,βγ,δ(z)including its auxiliary functions and the integral representations are proven.

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22

Milovanović, Gradimir V., Rakesh K. Parmar, and Arjun K. Rathie. "Certain Laplace transforms of convolution type integrals involving product of two special pFp functions." Demonstratio Mathematica 51, no. 1 (2018): 264–76. http://dx.doi.org/10.1515/dema-2018-0025.

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Abstract Recently the authors obtained several Laplace transforms of convolution type integrals involving Kummer’s function 1F1 [Appl. Anal. Discrete Math., 2018, 12(1), 257-272]. In this paper, the authors aim at presenting several new and interesting Laplace transforms of convolution type integrals involving product of two special generalized hypergeometric functions pFp by employing classical summation theorems for the series 2F1, 3F2, 4F3 and 5F4 available in the literature.

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23

Raj Tripathi, Bhadra. "Extension of Connecting Formulas on Hypergeometric Function." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 16, no. 1 (2025): 123–34. https://doi.org/10.61841/turcomat.v16i1.15233.

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The Hypergeometric series is the extension of the geometric series and the Confluent Hypergeometric Function is the solution of the Hypergeometric Differential Equation. Kummer has developed six solutions for the differential equation and twenty connecting formulas. The connecting formula consist of a solution expressed as the combination of two other solutions. Further extension was done by Poudel et al. This research work has extended the nine connecting formulas obtained by Poudel et al. to obtain the other nine solutions.

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Vyas, Yashoverdhan, Hari M. Srivastava, Shivani Pathak, and Kalpana Fatawat. "General Summation Formulas Contiguous to the q-Kummer Summation Theorems and Their Applications." Symmetry 13, no. 6 (2021): 1102. http://dx.doi.org/10.3390/sym13061102.

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This paper provides three classes of q-summation formulas in the form of general contiguous extensions of the first q-Kummer summation theorem. Their derivations are presented by using three methods, which are along the lines of the three types of well-known proofs of the q-Kummer summation theorem with a key role of the q-binomial theorem. In addition to the q-binomial theorem, the first proof makes use of Thomae’s q-integral representation and the second proof needs Heine’s transformation. Whereas the third proof utilizes only the q-binomial theorem. Subsequently, the applications of these summation formulas in obtaining the general contiguous extensions of the second and the third q-Kummer summation theorems are also presented. Furthermore, the investigated results are specialized to give many of the known as well as presumably new q-summation theorems, which are contiguous to the three q-Kummer summation theorems. This work is motivated by the observation that the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) gamma and q-hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas including Number Theory, Theory of Partitions and Combinatorial Analysis as well as in the study of Combinatorial Generating Functions. Just as it is known in the theory of the Gauss, Kummer (or confluent), Clausen and the generalized hypergeometric functions, the parameters in the corresponding basic or quantum (or q-) hypergeometric functions are symmetric in the sense that they remain invariant when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. A case has therefore been made for the symmetry possessed not only by hypergeometric functions and basic or quantum (or q-) hypergeometric functions, which are studied in this paper, but also by the symmetric quantum calculus itself.

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Hu, Jiaxin, Chenglong Yu, and Kangyun Zhou. "Padé Approximations and Irrationality Measures on Values of Confluent Hypergeometric Functions." Mathematics 12, no. 16 (2024): 2516. http://dx.doi.org/10.3390/math12162516.

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Padé approximations are approximations of holomorphic functions by rational functions. The application of Padé approximations to Diophantine approximations has a long history dating back to Hermite. In this paper, we use the Maier–Chudnovsky construction of Padé-type approximation to study irrationality properties about values of functions with the form f(x)=∑k=0∞xkk!(bk+s)(bk+s+1)⋯(bk+t), where b,t,s are positive integers and obtain upper bounds for irrationality measures of their values at nonzero rational points. Important examples includes exponential integral, Gauss error function and Kummer’s confluent hypergeometric functions.

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Atash, Ahmed Ali, and Hussein Saleh Bellehaj. "Applications of the Generalized Kummer’s Summation Theorem to Transformation Formulas and Generating Functions." Applied Mathematics and Nonlinear Sciences 3, no. 2 (2018): 331–38. http://dx.doi.org/10.21042/amns.2018.2.00026.

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AbstractIn this paper, we establish two general transformation formulas for Exton’s quadruple hypergeometric functions K5 and K12 by application of the generalized Kummer’s summation theorem. Further, a number of generating functions for Jacobi polynomials are also derived as an applications of our main results.

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27

Saxena, R. K., Chena Ram, Naresh Dudi, and S. L. Kalla. "Generalized gamma-type functions involving Kummer’s confluent hypergeometric function and associated probability distributions." Integral Transforms and Special Functions 18, no. 9 (2007): 679–87. http://dx.doi.org/10.1080/10652460701510501.

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28

Byrnes, Alyssa, Lin Jiu, Victor H. Moll, and Christophe Vignat. "Recursion rules for the hypergeometric zeta function." International Journal of Number Theory 10, no. 07 (2014): 1761–82. http://dx.doi.org/10.1142/s1793042114500547.

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The hypergeometric zeta function is defined in terms of the zeros of the Kummer function M(a, a+b; z). It is established that this function is an entire function of order 1. The classical factorization theorem of Hadamard gives an expression as an infinite product. This provides linear and quadratic recurrences for the hypergeometric zeta function. A family of associated polynomials is characterized as Appell polynomials and the underlying distribution is given explicitly in terms of the zeros of the associated hypergeometric function. These properties are also given a probabilistic interpretation in the framework of beta distributions.

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29

Kim, Yong Sup, Arjun K. Rathie, and Djurdje Cvijović. "New Laplace transforms of Kummer’s confluent hypergeometric functions." Mathematical and Computer Modelling 55, no. 3-4 (2012): 1068–71. http://dx.doi.org/10.1016/j.mcm.2011.09.031.

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30

Tarasov, Vasily, and Valentina Tarasova. "Dynamic Keynesian Model of Economic Growth with Memory and Lag." Mathematics 7, no. 2 (2019): 178. http://dx.doi.org/10.3390/math7020178.

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A mathematical model of economic growth with fading memory and continuous distribution of delay time is suggested. This model can be considered as a generalization of the standard Keynesian macroeconomic model. To take into account the memory and gamma-distributed lag we use the Abel-type integral and integro-differential operators with the confluent hypergeometric Kummer function in the kernel. These operators allow us to propose an economic accelerator, in which the memory and lag are taken into account. The fractional differential equation, which describes the dynamics of national income in this generalized model, is suggested. The solution of this fractional differential equation is obtained in the form of series of the confluent hypergeometric Kummer functions. The asymptotic behavior of national income, which is described by this solution, is considered.

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Kotlyar, V. V., A. A. Kovalev, and E. G. Abramochkin. "Asymmetric hypergeometric laser beams." Computer Optics 43, no. 5 (2019): 735–40. http://dx.doi.org/10.18287/2412-6179-2019-43-5-735-740.

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Here we study asymmetric Kummer beams (aK-beams) with their scalar complex amplitude being proportional to the Kummer function (a degenerate hypergeometric function). These beams are an exact solution of the paraxial propagation equation (Schrödinger-type equation) and obtained from the conventional symmetric hypergeometric beams by a complex shift of the transverse coordinates. On propagation, the aK-beams change their intensity weakly and rotate around the optical axis. These beams are an example of vortex laser beams with a fractional orbital angular momentum (OAM), which depends on four parameters: the vortex topological charge, the shift magnitude, the logarithmic axicon parameter and the degree of the radial factor. Changing these parameters, it is possible to control the beam OAM, either continuously increasing or decreasing it.

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32

Sobajima, Motohiro, and Yuta Wakasugi. "Weighted energy estimates for wave equation with space-dependent damping term for slowly decaying initial data." Communications in Contemporary Mathematics 21, no. 05 (2019): 1850035. http://dx.doi.org/10.1142/s0219199718500359.

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This paper is concerned with weighted energy estimates for solutions to wave equation [Formula: see text] with space-dependent damping term [Formula: see text] [Formula: see text] in an exterior domain [Formula: see text] having a smooth boundary. The main result asserts that the weighted energy estimates with weight function like polynomials are given and these decay rates are almost sharp, even when the initial data do not have compact support in [Formula: see text]. The crucial idea is to use special solution of [Formula: see text] including Kummer’s confluent hypergeometric functions.

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Pandey, Neelam, and Akanksha Srivastava. "Kummer and Dixon Summation Theorems: Applications in Hypergeometric Functions and Double Series." Journal of Advances and Scholarly Researches in Allied Education 21, no. 7 (2024): 5–11. https://doi.org/10.29070/yp9tjf09.

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The Kummer and Dixon summation theorems essentially carry out the essential task of simplifying hypergeometric functions and double series. This essay will cover both the topic and the basic concept of hypergeometric functions as well as the understanding and importance of summation theorems in mathematical analysis. We explore the theorems made by Kummer and Dixon, show how their theorems are relevant and explain them by using particular examples involving the hypergeometric series. The other topic considered also is regarding the convergence of double series and we explain how these summation theorems can be used to make evaluation of them simpler. The comparative study exhibits the interworking nature between Kummer and Dixon telegrams to describe instances where their compounded application is advantageous. The lecture goes further in the topic and scrutinizes the application of these functions in multivariable hypergeometric as well as mathematics and physics. The method is also discussed. Besides this, the latest headway and unsolved issues are also highlighted with some prospective research targets being listed for the future. The Noether’s Theorem illuminates the concise relation between the affinity of the Dixon and Bernoulli Summation Theorems to maths and physics, therefore dispelling any misconceptions about its practicality. Our research aim is to provide the necessary spark for the discovery and invention in this area by developing new approaches and methods which will eventually lead to the growth of mathematical research and its practical applications.

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34

Qureshi, M. I., M. Sadiq Khan, and M. A. Pathan. "Some multiple Gaussian hypergeometric generalizations of Buschman-Srivastava theorem." International Journal of Mathematics and Mathematical Sciences 2005, no. 1 (2005): 143–53. http://dx.doi.org/10.1155/ijmms.2005.143.

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Some generalizations of Bailey's theorem involving the product of two Kummer functions1F1are obtained by using Watson's theorem and Srivastava's identities. Its special cases yield various new transformations and reduction formulae involving Pathan's quadruple hypergeometric functionsFp(4), Srivastava's triple and quadruple hypergeometric functionsF(3),F(4), Lauricella's quadruple hypergeometric functionFA(4), Exton's multiple hypergeometric functionsXE:G;HA:B;D,K10,K13,X8,(k)H2(n),(k)H4(n), Erdélyi's multiple hypergeometric functionHn,k, Khan and Pathan's triple hypergeometric functionH4(P), Kampé de Fériet's double hypergeometric functionFE:G;HA:B;D, Appell's double hypergeometric function of the second kindF2, and the Srivastava-Daoust functionFD:E(1);E(2);…;E(n)A:B(1);B(2);…;B(n). Some known results of Buschman, Srivastava, and Bailey are obtained.

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35

Oros, Georgia Irina. "Applications of Inequalities in the Complex Plane Associated with Confluent Hypergeometric Function." Symmetry 13, no. 2 (2021): 259. http://dx.doi.org/10.3390/sym13020259.

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The idea of inequality has been extended from the real plane to the complex plane through the notion of subordination introduced by Professors Miller and Mocanu in two papers published in 1978 and 1981. With this notion came a whole new theory called the theory of differential subordination or admissible functions theory. Later, in 2003, a particular form of inequality in the complex plane was also defined by them as dual notion for subordination, the notion of differential superordination and with it, the theory of differential superordination appeared. In this paper, the theory of differential superordination is applied to confluent hypergeometric function. Hypergeometric functions are intensely studied nowadays, the interest on the applications of those functions in complex analysis being renewed by their use in the proof of Bieberbach’s conjecture given by de Branges in 1985. Using the theory of differential superodination, best subordinants of certain differential superordinations involving confluent (Kummer) hypergeometric function are stated in the theorems and relation with previously obtained results are highlighted in corollaries using particular functions and in a sandwich-type theorem. An example is also enclosed in order to show how the theoretical findings can be applied.

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36

Metzler, Adam. "The Laplace Transform of Hitting Times of Integrated Geometric Brownian Motion." Journal of Applied Probability 50, no. 1 (2013): 295–99. http://dx.doi.org/10.1239/jap/1363784440.

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In this note we compute the Laplace transform of hitting times, to fixed levels, of integrated geometric Brownian motion. The transform is expressed in terms of the gamma and confluent hypergeometric functions. Using a simple Itô transformation and standard results on hitting times of diffusion processes, the transform is characterized as the solution to a linear second-order ordinary differential equation which, modulo a change of variables, is equivalent to Kummer's equation.

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37

Metzler, Adam. "The Laplace Transform of Hitting Times of Integrated Geometric Brownian Motion." Journal of Applied Probability 50, no. 01 (2013): 295–99. http://dx.doi.org/10.1017/s0021900200013279.

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In this note we compute the Laplace transform of hitting times, to fixed levels, of integrated geometric Brownian motion. The transform is expressed in terms of the gamma and confluent hypergeometric functions. Using a simple Itô transformation and standard results on hitting times of diffusion processes, the transform is characterized as the solution to a linear second-order ordinary differential equation which, modulo a change of variables, is equivalent to Kummer's equation.

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38

Qi, Feng, and Bai-Ni Guo. "On the sum of the Lah numbers and zeros of the Kummer confluent hypergeometric function." Acta Universitatis Sapientiae, Mathematica 10, no. 1 (2018): 125–33. http://dx.doi.org/10.2478/ausm-2018-0011.

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39

Nagar, Daya K., Edwin Zarrazola, and Jessica Serna-Morales. "Generalized Bivariate Kummer-Beta Distribution." Ingeniería y Ciencia 16, no. 32 (2020): 7–31. http://dx.doi.org/10.17230/ingciencia.16.32.1.

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A new bivariate beta distribution based on the Humbert’s confluent hypergeometric function of the second kind is introduced. Various representations are derived for its product moments, marginal densities, marginal moments, conditional densities and entropies.

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40

Oros, Georgia Irina. "Fuzzy Differential Subordinations Obtained Using a Hypergeometric Integral Operator." Mathematics 9, no. 20 (2021): 2539. http://dx.doi.org/10.3390/math9202539.

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This paper is related to notions adapted from fuzzy set theory to the field of complex analysis, namely fuzzy differential subordinations. Using the ideas specific to geometric function theory from the field of complex analysis, fuzzy differential subordination results are obtained using a new integral operator introduced in this paper using the well-known confluent hypergeometric function, also known as the Kummer hypergeometric function. The new hypergeometric integral operator is defined by choosing particular parameters, having as inspiration the operator studied by Miller, Mocanu and Reade in 1978. Theorems are stated and proved, which give corollary conditions such that the newly-defined integral operator is starlike, convex and close-to-convex, respectively. The example given at the end of the paper proves the applicability of the obtained results.

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41

Nagar, Daya K., Edwin Zarrazola, and Alejandro Roldán-Correa. "Conditionally Specified Bivariate Kummer-Gamma Distribution." WSEAS TRANSACTIONS ON MATHEMATICS 20 (April 29, 2021): 196–206. http://dx.doi.org/10.37394/23206.2021.20.21.

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The Kummer-gamma distribution is an extension of gamma distribution and for certain values of parameters slides to a bimodal distribution. In this article, we introduce a bivariate distribution with Kummer-gamma conditionals and call it the conditionally specified bivariate Kummer-gamma distribution/bivariate Kummer-gamma conditionals distribution. Various representations are derived for its product moments, marginal densities, marginal moments, conditional densities, and conditional moments. We also discuss several important properties including, entropies, distributions of sum, and quotient. Most of these representations involve special functions such as the Gauss and the confluent hypergeometric functions. The bivariate Kummer-gamma conditionals distribution studied in this article may serve as an alternative to many existing bivariate models with support on (0, ∞) × (0, ∞).

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42

TERASOMA, Tomohide. "Exponential Kummer Coverings and Determinants of Hypergeometric Functions." Tokyo Journal of Mathematics 16, no. 2 (1993): 497–508. http://dx.doi.org/10.3836/tjm/1270128499.

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43

CHOI, JEONG-RYEOL. "WAVE FUNCTIONS WITH DISCRETE AND WITH CONTINUOUS SPECTRUM FOR QUANTUM DAMPED HARMONIC OSCILLATOR PERTURBED BY A SINGULARITY." International Journal of Modern Physics B 18, no. 07 (2004): 1007–20. http://dx.doi.org/10.1142/s0217979204024495.

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The quantum states with discrete and continuous spectrum for the damped harmonic oscillator perturbed by a singularity have been investigated using invariant operator and unitary operator together. The eigenvalue of the invariant operator for ω0≤β/2 is continuous while for ω0>β/2 is discrete. The wave functions for ω0=β/2 expressed in terms of the Bessel function and for ω0<β/2 in terms of the Kummer confluent hypergeometric function. The convergence of the probability density is more rapid for over-damped harmonic oscillator than that of the other two cases due to the large damping constant.

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44

Paris, R. B. "A Kummer-type transformation for a 2F2 hypergeometric function." Journal of Computational and Applied Mathematics 173, no. 2 (2005): 379–82. http://dx.doi.org/10.1016/j.cam.2004.05.005.

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45

Choi, Junesang, Gradimir V. Milovanović, and Arjun K. Rathie. "Generalized Summation Formulas for the KampÉ de FÉriet Function." Axioms 10, no. 4 (2021): 318. http://dx.doi.org/10.3390/axioms10040318.

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By employing two well-known Euler’s transformations for the hypergeometric function 2F1, Liu and Wang established numerous general transformation and reduction formulas for the Kampé de Fériet function and deduced many new summation formulas for the Kampé de Fériet function with the aid of classical summation theorems for the 2F1 due to Kummer, Gauss and Bailey. Here, by making a fundamental use of the above-mentioned reduction formulas, we aim to establish 32 general summation formulas for the Kampé de Fériet function with the help of generalizations of the above-referred summation formulas for the 2F1 due to Kummer, Gauss and Bailey. Relevant connections of some particular cases of our main identities, among numerous ones, with those known formulas are explicitly indicated.

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46

Hill, Adrian T. "Estimates on the Green's function of second-order elliptic operators in ℝN". Proceedings of the Royal Society of Edinburgh: Section A Mathematics 128, № 5 (1998): 1033–51. http://dx.doi.org/10.1017/s0308210500030055.

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Sharp upper and lower pointwise bounds are obtained for the Green's function of the equationfor λ> 0. Initially, in a Cartesian frame, it is assumed that . Pointwise estimates for the heat kernel of ut + Lu = 0, recently obtained under this assumption, are Laplace-transformed to yield corresponding elliptic results. In a second approach, the coordinate-free constraint is imposed. Within this class of operators, the equations defining the maximal and minimal Green's functions are found to be simple ODEs when written in polar coordinates, and these are soluble in terms of the singular Kummer confluent hypergeometric function. For both approaches, bounds on are derived as a consequence.

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47

Choi, Junesang. "Certain Applications of Generalized Kummer’s Summation Formulas for 2F1." Symmetry 13, no. 8 (2021): 1538. http://dx.doi.org/10.3390/sym13081538.

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We present generalizations of three classical summation formulas 2F1 due to Kummer, which are able to be derived from six known summation formulas of those types. As certain simple particular cases of the summation formulas provided here, we give a number of interesting formulas for double-finite series involving quotients of Gamma functions. We also consider several other applications of these formulas. Certain symmetries occur often in mathematical formulae and identities, both explicitly and implicitly. As an example, as mentioned in Remark 1, evident symmetries are naturally implicated in the treatment of generalized hypergeometric series.

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48

Kim, Insuk, and Arjun K. Rathie. "A Note on Certain General Transformation Formulas for the Appell and the Horn Functions." Symmetry 15, no. 3 (2023): 696. http://dx.doi.org/10.3390/sym15030696.

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In a number of problems in applied mathematics, physics (theoretical and mathematical), statistics, and other fields the hypergeometric functions of one and several variables naturally appear. Hypergeometric functions in one and several variables have several known applications today. The Appell’s four functions and the Horn’s functions have shown to be particularly useful in providing solutions to a variety of problems in both pure and applied mathematics. The Hubbell rectangular source and its generalization, non-relativistic theory, and the hydrogen dipole matrix elements are only a few examples of the numerous scientific and chemical domains where Appell functions are used. The Appell series is also used in quantum field theory, especially in the evaluation of Feynman integrals. Additionally, since 1985, computational sciences such as artificial intelligence (AI) and information processing (IP) have used the well-known Horn functions as a key idea. In literature, there have been published a significant number of results of double series in particular of Appell and Horn functions in a series of interesting and useful research publications. We find three general transformation formulas between Appell functions F2 and F4 and two general transformation formulas between Appell function F2 and Horn function H4 in the present study, which are mostly inspired by their work and naturally exhibit symmetry. By using the generalizations of the Kummer second theorem in the integral representation of the Appell function F2, this is accomplished. As special cases of our main findings, both previously known and new results have been found.

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49

Kella, Offer, and Wolfgang Stadje. "On hitting times for compound Poisson dams with exponential jumps and linear release rate." Journal of Applied Probability 38, no. 3 (2001): 781–86. http://dx.doi.org/10.1239/jap/1005091042.

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For a compound Poisson dam with exponential jumps and linear release rate (shot-noise process), we compute the Laplace-Stieltjes transform (LST) and the mean of the hitting time of some positive level given that the process starts from some given positive level. The solution for the LST is in terms of confluent hypergeometric functions of the first and second kinds (Kummer functions).

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50

Kella, Offer, and Wolfgang Stadje. "On hitting times for compound Poisson dams with exponential jumps and linear release rate." Journal of Applied Probability 38, no. 03 (2001): 781–86. http://dx.doi.org/10.1017/s0021900200018945.

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For a compound Poisson dam with exponential jumps and linear release rate (shot-noise process), we compute the Laplace-Stieltjes transform (LST) and the mean of the hitting time of some positive level given that the process starts from some given positive level. The solution for the LST is in terms of confluent hypergeometric functions of the first and second kinds (Kummer functions).

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