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Articles de revues sur le sujet « Q-series expansion »

Auteur : Grafiati

Publié le 2 juin 2025

Mis à jour le 31 juillet 2025

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1

McIntosh, Richard J. "Asymptotic Transformations of q-Series." Canadian Journal of Mathematics 50, no. 2 (1998): 412–25. http://dx.doi.org/10.4153/cjm-1998-022-x.

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AbstractFor the q–series we construct a companion q–series such that the asymptotic expansions of their logarithms as q → 1– differ only in the dominant few terms. The asymptotic expansion of their quotient then has a simple closed form; this gives rise to a new q–hypergeometric identity. We give an asymptotic expansion of a general class of q–series containing some of Ramanujan's mock theta functions and Selberg's identities.

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2

Annaby, M. H., Z. S. Mansour, and I. A. Soliman. "q-Titchmarsh-Weyl theory: series expansion." Nagoya Mathematical Journal 205 (March 2012): 67–118. http://dx.doi.org/10.1215/00277630-1543787.

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AbstractWe establish a q-Titchmarsh-Weyl theory for singular q-Sturm-Liouville problems. We define q-limit-point and q-limit circle singularities, and we give sufficient conditions which guarantee that the singular point is in a limit-point case. The resolvent is constructed in terms of Green’s function of the problem. We derive the eigenfunction expansion in its series form. A detailed worked example involving Jackson q-Bessel functions is given. This example leads to the completeness of a wide class of q-cylindrical functions.

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3

Annaby, M. H., Z. S. Mansour, and I. A. Soliman. "q-Titchmarsh-Weyl theory: series expansion." Nagoya Mathematical Journal 205 (March 2012): 67–118. http://dx.doi.org/10.1017/s002776300001045x.

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AbstractWe establish aq-Titchmarsh-Weyl theory for singularq-Sturm-Liouville problems. We defineq-limit-point andq-limit circle singularities, and we give sufficient conditions which guarantee that the singular point is in a limit-point case. The resolvent is constructed in terms of Green’s function of the problem. We derive the eigenfunction expansion in its series form. A detailed worked example involving Jacksonq-Bessel functions is given. This example leads to the completeness of a wide class ofq-cylindrical functions.

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4

Al-Towailb, Maryam, and Zeinab S. I. Mansour. "A q-Analog of the Class of Completely Convex Functions and Lidstone Series." Axioms 12, no. 5 (2023): 412. http://dx.doi.org/10.3390/axioms12050412.

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This paper introduces a q-analog of the class of completely convex functions. We prove specific properties, including that q-completely convex functions have convergent q-Lidstone series expansions. We also provide a sufficient and necessary condition for a real function to have an absolutely convergent q-Lidstone series expansion.

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5

LIU, ZHI-GUO. "ON THE q-DERIVATIVE AND q-SERIES EXPANSIONS." International Journal of Number Theory 09, no. 08 (2013): 2069–89. http://dx.doi.org/10.1142/s1793042113500759.

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Using a general q-series expansion, we derive some nontrivial q-formulas involving many infinite products. A multitude of Hecke-type series identities are derived. Some general formulas for sums of any number of squares are given. A new representation for the generating function for sums of three triangular numbers is derived, which is slightly different from that of Andrews, also implies the famous result of Gauss where every integer is the sum of three triangular numbers.

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Banerjee, Shubho, and Blake Wilkerson. "Asymptotic expansions of Lambert series and related q-series." International Journal of Number Theory 13, no. 08 (2017): 2097–113. http://dx.doi.org/10.1142/s1793042117501135.

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We study the Lambert series [Formula: see text], for all [Formula: see text]. We obtain the complete asymptotic expansion of [Formula: see text] near [Formula: see text]. Our analysis of the Lambert series yields the asymptotic forms for several related [Formula: see text]-series: the [Formula: see text]-gamma and [Formula: see text]-polygamma functions, the [Formula: see text]-Pochhammer symbol and the Jacobi theta functions. Some typical results include [Formula: see text] and [Formula: see text], with relative errors of order [Formula: see text] and [Formula: see text] respectively.

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7

Al-Towailb, Maryam, and Zeinab Mansour. "Conditional Expanding of Functions by q-Lidstone Series." Axioms 12, no. 1 (2022): 22. http://dx.doi.org/10.3390/axioms12010022.

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This paper characterizes those functions given by convergent q-Lidstone series expansion. We give the necessary and sufficient conditions so that the entire function f(z) has such an expansion, in which case convergence is uniform on compact sets.

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8

Ahmad, Naeem, and Waseem Ahmad Khan. "Insights into New Generalization of q-Legendre-Based Appell Polynomials: Properties and Quasi Monomiality." Mathematics 13, no. 6 (2025): 955. https://doi.org/10.3390/math13060955.

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In this paper, by using the zeroth-order q-Tricomi functions, the theory of three-variable q-Legendre-based Appell polynomials is introduced. These polynomials are studied by means of generating functions, series expansions, and determinant representation. Further, by utilizing the concepts of q-quasi-monomiality, these polynomials are examined as several q-quasi-monomial and operational representations; the q-differential equations for the three-variable q-Legendre-based Appell polynomials were obtained. In addition, we established a new generalization of three-variable q-Legendre-Hermite-App

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9

Charalambides, Charalambos A. "A Class of Power Series q-Distributions." Mathematics 12, no. 5 (2024): 712. http://dx.doi.org/10.3390/math12050712.

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A class of power series q-distributions, generated by considering a q-Taylor expansion of a parametric function into powers of the parameter, is discussed. Its q-factorial moments are obtained in terms of q-derivatives of its series (parametric) function. Also, it is shown that the convolution of power series q-distributions is also a power series q-distribution. Furthermore, the q-Poisson (Heine and Euler), q-binomial of the first kind, negative q-binomial of the second kind, and q-logarithmic distributions are shown to be members of this class of distributions and their q-factorial moments a

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10

He, Bing, and Suzhen Wen. "Some expansion formulas for q-series and their applications." Journal of Combinatorial Theory, Series A 209 (January 2025): 105941. http://dx.doi.org/10.1016/j.jcta.2024.105941.

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Caalim, Jonathan, and Shiela Demegillo. "BETA CANTOR SERIES EXPANSION AND ADMISSIBLE SEQUENCES." Acta Polytechnica 60, no. 3 (2020): 214–24. http://dx.doi.org/10.14311/ap.2020.60.0214.

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We introduce a numeration system, called the <em>beta Cantor series expansion</em>, that generalizes the classical positive and negative beta expansions by allowing non-integer bases in the Q-Cantor series expansion. In particular, we show that for a fix $\gamma \in \mathbb{R}$ and a sequence $B=\{\beta_i\}$ of real number bases, every element of the interval $x \in [\gamma,\gamma+1)$ has a <em>beta Cantor series expansion</em> with respect to B where the digits are integers in some alphabet $\mathcal{A}(B)$. We give a criterion in determining whether an integer sequenc

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12

HANSEL, D., and J. M. MAILLARD. "SERIES ANALYSIS OF Q-STATE CHECKERBOARD POTTS MODELS." International Journal of Modern Physics B 02, no. 06 (1988): 1447–62. http://dx.doi.org/10.1142/s0217979288001293.

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The series analysis of the low temperature expansion of the checkerboard q-state Potts model in a magnetic field initiated in two previous papers is continued. In particular algebraic varieties of the parameter space (corresponding or generalizing the so-called disorder solutions), the checkerboard Potts model and its Bethe approximation are indistinguishable as far as one is concerned with the partition function and its first order derivatives. The difference between the two models occurs for higher order derivatives. In particular one gives the exact expression of the (low temperature expans

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13

Muzaffar, Habib. "Some interesting series arising from the power series expansion of(sin-1x)q." International Journal of Mathematics and Mathematical Sciences 2005, no. 14 (2005): 2329–36. http://dx.doi.org/10.1155/ijmms.2005.2329.

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14

Mahjoub, Faïza. "Purely periodic β-expansions in cubic Salem base in Fq((x−1))". Publications de l'Institut Math?matique (Belgrade) 100, № 114 (2016): 279–85. http://dx.doi.org/10.2298/pim1614279m.

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Let Fq be the finite field with q elements and ? Salem series in Fq((X?1)). It is proved in [15] that, in this case, all elements in Fq(X,?) have purely periodic ?-expansion. We characterize the formal power series f in Fq(X,?) with purely periodic ?-expansions by the conjugate vector ~f when ? is a cubic unit. No similar results exist in the real case.

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15

Bakaev, A. V., A. N. Ermilov, V. I. Kabanovich, and A. M. Kurbatov. "l/(q-1)-Expansion study of the q-color problem on a lattice." Canadian Journal of Physics 67, no. 5 (1989): 497–500. http://dx.doi.org/10.1139/p89-090.

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16

Liu, Zhi-Guo. "A q-series expansion formula and the Askey–Wilson polynomials." Ramanujan Journal 30, no. 2 (2012): 193–210. http://dx.doi.org/10.1007/s11139-012-9450-2.

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17

AL-Towailb, Maryam. "A generalization of the $ q $-Lidstone series." AIMS Mathematics 7, no. 5 (2022): 9339–52. http://dx.doi.org/10.3934/math.2022518.

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<abstract><p>In this paper, we study the existence of solutions for the general $ q $-Lidstone problem:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} (D_{q^{-1}}^{r_n}f)(1) = a_n, \quad (D_{q^{-1}}^{s_n}f)(0) = b_n, \quad (n\in \mathbb{N}) \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ (r_n)_n $ and $ (s_n)_n $ are two sequences of non-negative integers and $ (a_n)_n $ and $ (b_n)_n $ are two sequences of complex numbers. We define a $ q^{-1} $-standard set of p

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18

Castro-García, M. P., S. Norniella-Llaneza, T. Fernández-González, Y. Centeno-Idiáñez, and T. Alonso-Sánchez. "Modeling Thermal Interference Between Shallow Geothermal Boreholes Using TRNSYS: Application to the Q-Thermie-Uniovi Boreholes (Asturias, Spain)." Applied Sciences 15, no. 10 (2025): 5591. https://doi.org/10.3390/app15105591.

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This study presents the implementation of a new Type for TRNSYS based on the integral exponential function, which is fundamental in subsurface heat transfer modeling. This function can be expressed as a series expansion, and the developed Type enables the representation of multiple terms in this expansion. To validate its accuracy, the model’s results are compared with experimental data from two geothermal boreholes located in Asturias, Spain: Q-Thermie-Uniovi 1 and Q-Thermie-Uniovi 2. These boreholes, spaced 7.3 m apart and each 48 m deep, were used in a thermal response test, where Q-Thermie

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19

Ghorbel, R. "Beta-expansion in Pisot and Salem unit bases in Fq ((x-1))." Filomat 38, no. 19 (2024): 6871–79. https://doi.org/10.2298/fil2419871g.

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In [11] K. Schmidt studied the lengths of periods occurring in the ?-expansion of a rational number r noted by Per? (r) for the Pisot numbers of a very special form satisfying ?2 = n? + 1 for some n ? 1. He showed the curious property ?Per? (p/q) = Per? (1/q)? for all positive integers p, q with p ? q = 1 and p < q. The aim of this paper is to prove that in the case of a formal power series over finite fields this property is true for special cubic Pisot unit basis.

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20

Mance, Bill. "Construction of normal numbers with respect to the Q-Cantor series expansion for certain Q." Acta Arithmetica 148, no. 2 (2011): 135–52. http://dx.doi.org/10.4064/aa148-2-3.

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21

Doyle, Greg, and Kenneth S. Williams. "Evaluation of some q-integrals in terms of the Dedekind eta function." Analysis 38, no. 2 (2018): 63–79. http://dx.doi.org/10.1515/anly-2018-0002.

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Abstract A q-integral is a definite integral of a function of q having an expansion in non-negative powers of q for {|q|<1} (q-series). In his book on hypergeometric series, N. J. Fine [N. J. Fine, Basic Hypergeometric Series and Applications, Math. Surveys Monogr. 27, American Mathematical Society, Providence, 1988] explicitly evaluated three q-integrals. For example, he showed that \int_{0}^{e^{-\pi}}\prod_{n=1}^{\infty}\frac{(1-q^{2n})^{20}}{(1-q^{n})^{16}}% dq=\frac{1}{16}. In this paper, we prove a general theorem which allows us to determine a wide class of integrals of this type. Thi

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22

Mansour, Zeinab S. I., and Maryam Al-Towailb. "A $ q $-Type $ k $-Lidstone series for entire functions." AIMS Mathematics 8, no. 6 (2023): 13525–42. http://dx.doi.org/10.3934/math.2023686.

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<abstract><p>In this paper, we consider the $ q $-type $ k $-Lidstone series. The series follows from expanding certain classes of entire functions in terms of Jackson $ q^{-1} $- derivatives at integers congruent to r modulo k, where $ k $ is a positive integer. We study the main properties of the fundamental polynomials that appear in the series expansion. We include a detailed study for the case $ k = 3 $ with some examples.</p></abstract>

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23

MC LAUGHLIN, JAMES. "FURTHER RESULTS ON VANISHING COEFFICIENTS IN INFINITE PRODUCT EXPANSIONS." Journal of the Australian Mathematical Society 98, no. 1 (2014): 69–77. http://dx.doi.org/10.1017/s1446788714000536.

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AbstractWe extend results of Andrews and Bressoud [‘Vanishing coefficients in infinite product expansions’, J. Aust. Math. Soc. Ser. A27(2) (1979), 199–202] on the vanishing of coefficients in the series expansions of certain infinite products. These results have the form that if $$\begin{eqnarray}\frac{(q^{r-tk},q^{mk-(r-tk)};q^{mk})_{\infty }}{(\pm q^{r},\pm q^{mk-r};q^{mk})_{\infty }}=:\mathop{\sum }_{n=0}^{\infty }c_{n}q^{n}\end{eqnarray}$$ for certain integers $k$, $m$, $s$ and $t$, where $r=sm+t$, then $c_{kn-rs}$ is always zero. Our theorems also partly give a simpler reformulation of r

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24

KIM, MIN-SOO, and SU HU. "A p-ADIC VIEW OF MULTIPLE SUMS OF POWERS." International Journal of Number Theory 07, no. 08 (2011): 2273–88. http://dx.doi.org/10.1142/s1793042111005027.

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Washington [p-Adic L-functions and sums of powers, J. Number Theory69 (1998) 50–61] gave an explicit p-adic expansion of [Formula: see text] as a power series in m. The coefficients are values of p-adic L-functions. Let q = 4 if p = 2 and q = p otherwise. In this paper, we prove an explicit p-adic expansion of the multiple sums of powers [Formula: see text] as a p-adic power series in m. The coefficients are values of multiple two-variable p-adic L-functions. Washington's formula is a special case of the formula given in this paper when n = 1 and t = 0.

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25

CHRISTE, P., and M. J. MARTINS. "ULTRAVIOLET BEHAVIOR OF THE INTEGRABLE OFF-CRITICAL NON-UNITARY SERIES M3/q." Modern Physics Letters A 05, no. 26 (1990): 2189–95. http://dx.doi.org/10.1142/s0217732390002493.

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We discuss the ultraviolet behavior of the non-unitary M3/q (q = 5, 7) minimal models perturbed by the operator ϕ13. The Thermodynamic Bethe Ansatz is used to compute the central charge and the next higher order corrections in the high temperature expansion of the finite volume ground state energy. The results are in perfect agreement with perturbative calculations. Generalizations of these results are commented on.

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26

Li, Xue, and Chao Ma. "The Shrinking Target Problem for Nonautonomous Dynamical Systems Corresponding to Cantor Series Expansion over Formal Laurent Series." Mathematics 12, no. 20 (2024): 3166. http://dx.doi.org/10.3390/math12203166.

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In this paper, we study the shrinking target problem regarding Q-Cantor series expansions of the formal Laurent series field. We provide the Hausdorff dimension of a very general shrinking target scheme generated by the nonautonomous dynamical system on the unit disk I.

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27

GOUBI, M. "Series Expansion of a Cotangent Sum Related to the Estermann Zeta Function." Kragujevac Journal of Mathematics 45, no. 03 (2021): 343–52. http://dx.doi.org/10.46793/kgjmat2103.343g.

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Ernst, Thomas. "On Various Formulas with q-integralsand Their Applications to q-hypergeometric Functions." European Journal of Pure and Applied Mathematics 13, no. 5 (2020): 1241–59. http://dx.doi.org/10.29020/nybg.ejpam.v13i5.3755.

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We present three q-Taylor formulas with q-integral remainder. The two last proofsrequire a slight rearrangement by a well-known formula. The first formula has been given in different form by Annaby and Mansour. We give concise proofs for q-analogues of Eulerian integral formulas for general q-hypergeometric functions corresponding to Erd ́elyi, and for two of Srivastavas triple hypergeometric functions and other functions. All proofs are made in a similar style by using q-integration. We find some new formulas for fractional q-integrals including a series expansion. In the same way, the operat

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29

Khan, Mohammad Faisal, та Mohammed AbaOud. "Some New Applications of the Faber Polynomial Expansion Method for Generalized Bi-Subordinate Functions of Complex Order γ Defined by q-Calculus". Fractal and Fractional 7, № 3 (2023): 270. http://dx.doi.org/10.3390/fractalfract7030270.

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This work examines a new subclass of generalized bi-subordinate functions of complex order γ connected to the q-difference operator. We obtain the upper bounds ρm for generalized bi-subordinate functions of complex order γ using the Faber polynomial expansion technique. Additionally, we find coefficient bounds ρ2 and Feke–Sezgo problems ρ3−ρ22 for the functions in the newly defined class, subject to gap series conditions. Using the Faber polynomial expansion method, we show some results that illustrate diverse uses of the Ruschewey q differential operator. The findings in this paper generalize

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Kuznetsov, Dmitriy Feliksovich. "Mean-square Approximation of Iterated Ito and Stratonovich Stochastic Integrals: Method of Generalized Multiple Fourier Series. Application to Numerical Integration of Ito Sdes and Semilinear Spdes (third Edition)." Differential Equations and Control Processes, no. 1 (2023): 151–1097. http://dx.doi.org/10.21638/11701/spbu35.2023.110.

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This is the third edition of the monograph (first edition 2020, second edition 2021) devoted to the problem of mean-square approximation of iterated Ito and Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. The mentioned problem is considered in the book as applied to the numerical integration of non-commutative Ito stochastic differential equations and semilinear stochastic partial differential equations with nonlinear non-commutative trace class noise. The book opens up a new direction in researching of iterated stochastic integrals. For the

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31

Ahmad, Naeem, and Waseem Ahmad Khan. "A New Generalization of q-Laguerre-Based Appell Polynomials and Quasi-Monomiality." Symmetry 17, no. 3 (2025): 439. https://doi.org/10.3390/sym17030439.

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In this paper, we define a new generalization of three-variable q-Laguerre polynomials and derive some properties. By using these polynomials, we introduce a new generalization of three-variable q-Laguerre-based Appell polynomials (3VqLbAP) through a generating function approach involving zeroth-order q-Bessel–Tricomi functions. These polynomials are studied by means of generating function, series expansion, and determinant representation. Also, these polynomials are further examined within the framework of q-quasi-monomiality, leading to the establishment of essential operational identities.

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Garoufalidis, Stavros, Jie Gu, and Marcos Mariño. "The Resurgent Structure of Quantum Knot Invariants." Communications in Mathematical Physics 386, no. 1 (2021): 469–93. http://dx.doi.org/10.1007/s00220-021-04076-0.

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AbstractThe asymptotic expansion of quantum knot invariants in complex Chern–Simons theory gives rise to factorially divergent formal power series. We conjecture that these series are resurgent functions whose Stokes automorphism is given by a pair of matrices of q-series with integer coefficients, which are determined explicitly by the fundamental solutions of a pair of linear q-difference equations. We further conjecture that for a hyperbolic knot, a distinguished entry of those matrices equals to the Dimofte–Gaiotto–Gukov 3D-index, and thus is given by a counting of BPS states. We illustrat

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Swarup, Chetan. "Certain New Applications of Faber Polynomial Expansion for a New Class of bi-Univalent Functions Associated with Symmetric q-Calculus." Symmetry 15, no. 7 (2023): 1407. http://dx.doi.org/10.3390/sym15071407.

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In this study, we applied the ideas of subordination and the symmetric q-difference operator and then defined the novel class of bi-univalent functions of complex order γ. We used the Faber polynomial expansion method to determine the upper bounds for the functions belonging to the newly defined class of complex order γ. For the functions in the newly specified class, we further obtained coefficient bounds ρ2 and the Fekete–Szegő problem ρ3−ρ22, both of which have been restricted by gap series. We demonstrate many applications of the symmetric Sălăgean q-differential operator using the Faber p

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Tamang, Bal Bahadur, and Ajay Singh. "The Solvability of Polynomial Pell’s Equation." Journal of Institute of Science and Technology 25, no. 2 (2020): 125–32. http://dx.doi.org/10.3126/jist.v25i2.33749.

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This article attempts to describe the continued fraction expansion of ÖD viewed as a Laurent series x-1. As the behavior of the continued fraction expansion of ÖD is related to the solvability of the polynomial Pell’s equation p2-Dq2=1 where D=f2+2g is monic quadratic polynomial with deg g<deg f and the solutions p, q must be integer polynomials. It gives a non-trivial solution if and only if the continued fraction expansion of ÖD is periodic.

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Kaper, Hans G., and Man Kam Kwong. "Asymptotics of the Titchmarsh-Weyl m-coefficient for integrable potentials." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 103, no. 3-4 (1986): 347–58. http://dx.doi.org/10.1017/s0308210500018990.

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This article is concerned with the asymptotic behaviour of m(λ), the Titchmarsh-Weyl m-coefficient, for the singular eigenvalue equation y“ + (λ − q(x))y = 0 on [0, ∞), as λ →∞ in a sector in the upper half of the complex plane. It is assumed that the potential function q is integrable near 0. A simplified proof is given of a result of Atkinson [7], who derived the first two terms in the asymptotic expansion of m(λ), and a sharper error bound is obtained. Theproof is then generalised to derive subsequent terms in the asymptotic expansion. It is shown that the Titchmarsh-Weyl m-coefficient admi

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36

Narita, Hiro-aki. "Fourier–Jacobi expansion of automorphic forms on Sp(1,q) generating quaternionic discrete series." Journal of Functional Analysis 239, no. 2 (2006): 638–82. http://dx.doi.org/10.1016/j.jfa.2006.03.015.

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Dargys, Adolfas, and Artūras Acus. "Exponentials of general multivector in 3D Clifford algebras." Nonlinear Analysis: Modelling and Control 27, no. 1 (2022): 179–97. http://dx.doi.org/10.15388/namc.2022.27.24476.

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Closed form expressions to calculate the exponential of a general multivector (MV) in Clifford geometric algebras (GAs) Clp;q are presented for n = p + q = 3. The obtained exponential formulas were applied to find exact GA trigonometric and hyperbolic functions of MV argument. We have verified that the presented exact formulas are in accord with series expansion of MV hyperbolic and trigonometric functions. The exponentials may be applied to solve GA differential equations, in signal and image processing, automatic control and robotics.

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SADEK, MOHAMMAD. "FORMAL GROUPS AND INVARIANT DIFFERENTIALS OF ELLIPTIC CURVES." Bulletin of the Australian Mathematical Society 92, no. 1 (2015): 44–51. http://dx.doi.org/10.1017/s0004972715000295.

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In this paper, we find a power series expansion of the invariant differential ${\it\omega}_{E}$ of an elliptic curve $E$ defined over $\mathbb{Q}$, where $E$ is described by certain families of Weierstrass equations. In addition, we derive several congruence relations satisfied by the trace of the Frobenius endomorphism of $E$.

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POOR, CRIS, JERRY SHURMAN, and DAVID S. YUEN. "THETA BLOCK FOURIER EXPANSIONS, BORCHERDS PRODUCTS AND A SEQUENCE OF NEWMAN AND SHANKS." Bulletin of the Australian Mathematical Society 98, no. 1 (2018): 48–59. http://dx.doi.org/10.1017/s000497271800031x.

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The ‘Borcherds products everywhere’ construction [Gritsenko et al., ‘Borcherds products everywhere’, J. Number Theory148 (2015), 164–195] creates paramodular Borcherds products from certain theta blocks. We prove that the $q$-order of every such Borcherds product lies in a sequence $\{C_{\unicode[STIX]{x1D708}}\}$, depending only on the $q$-order $\unicode[STIX]{x1D708}$ of the theta block. Similarly, the $q$-order of the leading Fourier–Jacobi coefficient of every such Borcherds product lies in a sequence $\{A_{\unicode[STIX]{x1D708}}\}$, and this is the sequence $\{a_{n}\}$ from work of Newm

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40

Hajij, Mustafa. "The Bubble skein element and applications." Journal of Knot Theory and Its Ramifications 23, no. 14 (2014): 1450076. http://dx.doi.org/10.1142/s021821651450076x.

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We study a certain skein element in the relative Kauffman bracket skein module of the disk with some marked points, and expand this element in terms of linearly independent elements of this module. This expansion is used to compute and study the head and the tail of the colored Jones polynomial and in particular we give a simple q-series for the tail of the knot 85. Furthermore, we use this expansion to obtain an easy determination of the theta coefficients.

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Ciolli, Fabio, and Francesco Fidaleo. "On the Thermodynamics of the q-Particles." Entropy 24, no. 2 (2022): 159. http://dx.doi.org/10.3390/e24020159.

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Since the grand partition function Zq for the so-called q-particles (i.e., quons), q∈(−1,1), cannot be computed by using the standard 2nd quantisation technique involving the full Fock space construction for q=0, and its q-deformations for the remaining cases, we determine such grand partition functions in order to obtain the natural generalisation of the Plank distribution to q∈[−1,1]. We also note the (non) surprising fact that the right grand partition function concerning the Boltzmann case (i.e., q=0) can be easily obtained by using the full Fock space 2nd quantisation, by considering the

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42

Szalay, Alexander S. "Constraints on the Biasing of Density Fluctuations." Symposium - International Astronomical Union 130 (1988): 163–67. http://dx.doi.org/10.1017/s0074180900135971.

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A general nonlinear function G(X) describing the biasing of primordial Gaussian density fluctuations is considered. Arbitrary N-point correlations of the biased field are calculated in the form of a series expansion in terms of the correlations of the Gaussian field. The observed scaling of the three point correlations in the galaxy distribution is satisfied, but the scaling coefficient Q has a nontrivial value Q = J2/J12, where Jk is the k-th term in the Hermite expansion of G(X). The three point function is always accompanied by a cubic term Q3ξ1ξ2ξ3, independent of the functional form of th

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Morton, H. R. "The coloured Jones function and Alexander polynomial for torus knots." Mathematical Proceedings of the Cambridge Philosophical Society 117, no. 1 (1995): 129–35. http://dx.doi.org/10.1017/s0305004100072959.

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AbstractIn [2] it was conjectured that the coloured Jones function of a framed knot K, or equivalently the Jones polynomials of all parallels of K, is sufficient to determine the Alexander polynomial of K. An explicit formula was proposed in terms of the power series expansion , where JK, k(h) is the SU(2)q quantum invariant of K when coloured by the irreducible module of dimension k, and q = eh is the quantum group parameter.In this paper I show that the explicit formula does give the Alexander polynomial when K is any torus knot.

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Dahms, M. "The iterative series-expansion method for quantitative texture analysis. II. Applications." Journal of Applied Crystallography 25, no. 2 (1992): 259–67. http://dx.doi.org/10.1107/s0021889891012499.

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The orientation distribution function (ODF) of the crystallites of polycrystalline materials can be calculated from experimentally measured pole density functions (pole figures). This procedure, called pole-figure inversion, can be achieved by the series-expansion method (harmonic method). As a consequence of the (hkl)-({\bar h}{\bar k}{\bar l}) superposition, the solution is mathematically not unique. There is a range of possible solutions (the kernel) that is only limited by the positivity condition of the distribution function. The complete distribution function f(g) can be split into two p

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Alegri, Mateus. "Identities involving sum of divisors, integer partitions and compositions." Online Journal of Analytic Combinatorics, no. 17 (December 31, 2022): 1–29. https://doi.org/10.61091/ojac-1703.

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In this paper we show some identities come from the \( q \)-identities of Euler, Jacobi, Gauss, and Rogers-Ramanujan. Some of these identities relate the function sum of divisors of a positive integer \( n \) and the number of integer partitions of \( n \). One of the most intriguing results found here is given by the next equation, for \( n > 0 \), \[ \sum_{l=1}^n \frac{1}{l!} \sum_{w_1+w_2+\cdots+w_l \in C(n)} \frac{\sigma_1(w_1) \sigma_1(w_2) \cdots \sigma_1(w_l)}{w_1 w_2 \cdots w_l} = p_1(n), \] where \( \sigma_1(n) \) is the sum of all positive divisors of \( n \), \( p_1(n) \) is the

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Lertchoosakul, Poj, and Radhakrishnan Nair. "On the Quantitative Metric Theory of Continued Fractions in Positive Characteristic." Proceedings of the Edinburgh Mathematical Society 61, no. 1 (2018): 283–93. http://dx.doi.org/10.1017/s0013091517000177.

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AbstractLet 𝔽q be the finite field of q elements. An analogue of the regular continued fraction expansion for an element α in the field of formal Laurent series over 𝔽q is given uniquely by $$\alpha = A_0(\alpha ) + \displaystyle{1 \over {A_1(\alpha ) + \displaystyle{1 \over {A_2(\alpha ) + \ddots }}}},$$ where $(A_{n}(\alpha))_{n=0}^{\infty}$ is a sequence of polynomials with coefficients in 𝔽q such that deg(An(α)) ⩾ 1 for all n ⩾ 1. In this paper, we provide quantitative versions of metrical results regarding averages of partial quotients. A sample result we prove is that, given any ϵ > 0

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Tsai, Shan-Ho. "Large-q series expansion for the ground-state degeneracy of the q-state Potts antiferromagnet on the (3⋅12^{2}) lattice." Physical Review E 57, no. 3 (1998): 2686–89. http://dx.doi.org/10.1103/physreve.57.2686.

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Amourah, Ala, Mohammad Al-Ityan, Abdullah Alsoboh, Mohammad Bani Raba'a, and Suha Hammad. "Fekete-Szeg¨o Inequalities for a New Class of Bi-Univalent Functions Defined via the Mittag-Leffler Function." European Journal of Pure and Applied Mathematics 18, no. 2 (2025): 6064. https://doi.org/10.29020/nybg.ejpam.v18i2.6064.

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In this paper, we introduce a new subclass of analytic functions denoted byMp,qΣ (∝, β), where we use the subordination relationship between the Mittag-Leffler function and the (p, q)-derivative of F(z) to define this new class. By employing the Taylor-Maclaurin series expansion, we focus on estimating the bounds for the coefficients |a2| and |a3|. Moreover, we establish Fekete Szeg¨o inequalities for functions within this class.

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Jin, Zhang, Wang Yan-Guo, Zhang Guo-Shu, Lan Hui-Tian, Zhang Hua, and Hao Ya-Ju. "Q estimation using multifrequency point average method based on the Taylor series expansion with a different order." Applied Geophysics 18, no. 4 (2021): 557–68. http://dx.doi.org/10.1007/s11770-021-0918-y.

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Nurali, Nurdiana Binti, and Aini Janteng. "Results on Toeplitz Determinants for Subclasses of Analytic Functions Associated to q-Derivative Operator." Science and Technology Indonesia 9, no. 2 (2024): 354–58. http://dx.doi.org/10.26554/sti.2024.9.2.354-358.

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An analytic function, also known as a holomorphic function, is a complex-valued function that is differentiable at every point within a given domain. In other words, a function f (z) is analytic in a domain U if it has a derivative f′(z) at every point z in U. Let A represent the set of functions f that are analytic within the open unit disk D = {z ∈ ℂ : |z| < 1}. These functions possess a normalized Taylor-Maclaurin series expansion written in the form f (z) = z + Í∞ n=2 an z n where an ∈ ℂ, n = 2, 3, . . .. In recent years, the field of q-calculus has gained significant attention and rese

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