Computational Aspects of Three-Term Recurrence Relations

Abstract

Previous article Next article Computational Aspects of Three-Term Recurrence RelationsWalter GautschiWalter Gautschihttps://doi.org/10.1137/1009002PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] M. Abramowitz, Review 58, Math. Tables Aids Comput, 10 (1956), 176– Google Scholar[2] Renata Babušková, Über numerische Stabilität einiger Rekursionsformeln, Apl. Mat., 9 (1964), 186–193 MR0187365 0123.11104 Google Scholar[3] T. A. Bancroft, Some recurrence formulae in the incomplete beta function ratio, Ann. Math. Statistics, 20 (1949), 451–455 MR0030653 0041.03402 CrossrefISIGoogle Scholar[4] G. Blanch, Numerical evaluation of continued fractions, SIAM Rev., 6 (1964), 383–421 10.1137/1006092 MR0171374 0133.38904 LinkISIGoogle Scholar[5] British Association For The Advancement Of Science, Bessel functions. Part II. Functions of positive integer order, British Association for the Advancement of Science, Mathematical Tables, vol. X, University Press, Cambridge, 1952xl+255 MR0050973 0049.09409 Google Scholar[6] Herbert Buchholz, Die konfluente hypergeometrische Funktion mit besonderer Berücksichtigung ihrer Anwendungen, Ergebnisse der angewandten Mathematik. Bd. 2, Springer-Verlag, Berlin, 1953xvi+234 MR0054783 0050.07402 CrossrefGoogle Scholar[7] C. W. Clenshaw, The numerical solution of linear differential equations in Chebyshev series, Proc. Cambridge Philos. Soc., 53 (1957), 134–149 MR0082196 0077.32503 CrossrefGoogle Scholar[8] Lothar Collatz, The numerical treatment of differential equations. 3d ed, Translated from a supplemented version of the 2d German edition by P. G. Williams. Die Grundlehren der mathematischen Wissenschaften, Bd. 60, Springer-Verlag, Berlin, 1960xv+568 pp. (1 plate) MR0109436 0086.32601 CrossrefGoogle Scholar[9] F. J. Corbató, On the computation of auxiliary functions for two-center integrals by means of a high-speed computer, J. Chem. Phys., 24 (1956), 452–453 CrossrefGoogle Scholar[10] Fernando J. Corbató and , Jack L. Uretsky, Generation of spherical Bessel functions in digital computers, J. Assoc. Comput. Mach., 6 (1959), 366–375 MR0105792 0087.28104 CrossrefISIGoogle Scholar[11] A. Erdélyi, Asymptotic expansions, Dover Publications Inc., New York, 1956vi+108 MR0078494 0070.29002 Google Scholar[12] Arthur Erdélyi, , Wilhelm Magnus, , Fritz Oberhettinger and , Francesco G. Tricomi, Higher transcendental functions. Vol. I, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953xxvi+302 MR0058756 0051.30303 Google Scholar[13] Arthur Erdélyi, , Wilhelm Magnus, , Fritz Oberhettinger and , Francesco G. Tricomi, Higher transcendental functions. Vol. II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953xvii+396 MR0058756 0052.29502 Google Scholar[14] M. A. Efgrafov, A new proof of Perron's theorem, Izvestiya Akad. Nauk SSSR. Ser. Mat., 17 (1953), 77–82 MR0058109 Google Scholar[15] L. Fox, A short table for Bessel functions of integer orders and large arguments, Royal Society Shorter Mathematical Tables, No. 3, Cambridge, at the University Press, 1954, 28– MR0065245 0058.11402 Google Scholar[16] Walter Gautschi, Recursive computation of the repeated integrals of the error function, Math. Comp., 15 (1961), 227–232 MR0136074 0099.11401 Google Scholar[17] W. Gautschi, Algorithm 222—Incomplete beta function ratios, Comm. ACM, 7 (1964), 143–144, Certification of Algorithm 222, Ibid., p. 244 10.1145/363958.363967 CrossrefISIGoogle Scholar[18] W. Gautschi, Algorithm 236—Bessel functions of the first kind, Comm. ACM, 7 (1964), 479–480, Certification of Algorithm 236, Ibid., 8 (1965), pp. 105–106 10.1145/355586.355587 CrossrefISIGoogle Scholar[19] W. Gautschi, Algorithm 259—Legendre functions for arguments larger than one, Comm. ACM, 8 (1965), 488–492 10.1145/365474.355591 CrossrefISIGoogle Scholar[20] W. Gautschi, Algorithm 292—Regular Coulomb wave functions, Comm. ACM, 9 (1966), 793–795 10.1145/365876.365884 CrossrefISIGoogle Scholar[21] A. O. Gel'Fond, Calculus of Finite Differences, Gosud. Izdat. Fiz.-Mat. Lit., Moscow, 1959, (German and French translations of the first edition are available.) Google Scholar[22] M. Goldstein and , R. M. Thaler, Recurrence techniques for the calculation of Bessel functions, Math. Tables Aids Comput., 13 (1959), 102–108 MR0105794 0092.33002 CrossrefGoogle Scholar[23] Marion C. Gray, Bessel functions of integral order and complex argument, Comm. ACM, 4 (1961), 169– 10.1145/355578.366318 MR0151652 CrossrefISIGoogle Scholar[24] Philip Hartman, Ordinary differential equations, John Wiley & Sons Inc., New York, 1964xiv+612 MR0171038 0125.32102 Google Scholar[25] D. R. Hartree, Some properties and applications of the repeated integrals of the error function, Mem. Proc. Manchester Lit. Philos. Soc., 80 (1936), 85–102 Google Scholar[26] Sin Hitotumatu, Note on the computation of Bessel functions through recurrence formula, J. Math. Soc. Japan, 15 (1963), 353–359 MR0156008 0131.34401 CrossrefGoogle Scholar[27] C. W. Jones, A short table for the Bessel functions In+1/2(x),2πKn+1/2(x), Cambridge, at the University Press, 1952, 20–, Royal Society Shorter Mathematical Tables, No. 1 MR0051573 0047.37203 Google Scholar[28] W. Kahan, Note on bounds for generating Bessel functions by recurrence, un published Google Scholar[29] P. Kreuser, Masters Thesis, Über das Verhalten der Integrale homogener linearer Differenzengleichungen im Unendlichen, Dissertation, University of Tübingen, Leipzig, 1914 Google Scholar[30] N. A. Logan, Survey of some early studies of the scattering of plane waves by a sphere, Proc. IEEE, 53 (1965), 773–785 CrossrefISIGoogle Scholar[31] A. V. Luk'Yanov, , I. B. Teplov and , M. K. Akimov, Tablicy volnovih kulonovskih funkciinodoť, Izdat. Akad. Nauk SSSR., Moscow, 1961, (English translation by D. E. Brown, Pergamon Press, New York, 1965) Google Scholar[32] Saburo Makinouchi, Note on the recurrence techniques for the calculation of Bessel functions $J\sb{v}(x)$, Tech. Rep. Osaka Univ., 16 (1965), 185–201 MR0198658 Google Scholar[33] Saburo Makinouchi, Note on the recurrence techniques for the calculation of Bessel functions $I\sb{v}(x)$, Tech. Rep. Osaka Univ., 16 (1965), 203–216 MR0198659 Google Scholar[34] Herbert Meschkowski, Differenzengleichungen, Studia Mathematica, Bd. XIV, Vandenhoeck & Ruprecht, Göttingen, 1959, 243– MR0109264 0084.29202 Google Scholar[35] L. M. Milne-Thomson, The Calculus of Finite Differences, Macmillan, London, 1933 0008.01801 Google Scholar[36] Google Scholar[37] N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer, Berlin, 1924 CrossrefGoogle Scholar[38] F. W. J. Olver, Error analysis of Miller's recurrence algorithm, Math. Comp., 18 (1964), 65–74 MR0169406 0115.34502 ISIGoogle Scholar[39] F. W. J. Olver, Bessel functions of integer order, Handbook of Mathematical Functions, NBS Appl. Math. Ser. 55, 1964, Chap. 9 Google Scholar[40] Morio Onoe, Tables of modified quotients of Bessel functions of the first kind for real and imaginary arguments, Columbia University Press, New York, 1958vii+338 MR0095590 0085.12301 Google Scholar[41] O. Perron, Über einen Satz des Herrn Poincaré, J. Reine Angew. Math., 136 (1909), 17–37 CrossrefGoogle Scholar[42] O. Perron, Über lineare Differenzengleichungen, Acta Math., 34 (1911), 109–137 CrossrefGoogle Scholar[43A] Oskar Perron, Die Lehre von den Kettenbrüchen. Bd I. Elementare Kettenbrüche, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1954vi+194 MR0064172 0056.05901 Google Scholar[43B] Oskar Perron, Die Lehre von den Kettenbrüchen. Dritte, verbesserte und erweiterte Aufl. Bd. II. Analytisch-funktionentheoretische Kettenbrüche, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1957vi+316 MR0085349 0077.06602 Google Scholar[44] S. Pincherle, Sur la génération de systèmes récurrents au moyen d'une équation linéaire différentielle, Acta Math., 16 (1892), 341–363 CrossrefGoogle Scholar[45] S. Pincherle, Delle funzioni ipergeometriche e di varie questioni ad esse attinenti, Giorn. Mat. Battaglini, 32 (1894), 209–291, esp. Ch. III. 15. (Also in: Opere Scelte, vol. 1, pp. 273–357.) Google Scholar[46] H. Poincare, Sur les Equations Lineaires aux Differentielles Ordinaires et aux Differences Finies, Amer. J. Math., 7 (1885), 203–258, (Also in: Oeuvres Henri Poincaré, vol. 1, pp. 226–289.) MR1505385 CrossrefGoogle Scholar[47] J. B. Randels and , R. F. Reeves, Note on empirical bounds for generating Bessel functions, Comm. ACM, 1 (1958), 3–5 10.1145/368819.368846 0087.28103 CrossrefISIGoogle Scholar[48] Lord (J. W. Strutt) Rayleigh, The incidence of light upon a transparent sphere of dimensions comparable with the wave-length, Proc. Roy. Soc. London Ser. A, 84 (1910), 25–46, (Also in: Scientific Papers by John William Strutt, Baron Rayleigh, vol. 5, pp. 547–568.) CrossrefGoogle Scholar[49] Louis Robin, Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome II, Gauthier-Villars, Paris, 1958viii+384 MR0101928 0088.05102 Google Scholar[50] A. Rotenberg, The calculation of toroidal harmonics, Math. Comput., 14 (1960), 274–276 MR0115264 0096.32205 CrossrefGoogle Scholar[51] F. W. Schäfke, Lösungstypen von Differenzengleichungen und Summengleichungen in normierten abelschen Gruppen, Math. Z., 88 (1965), 61–104 MR0176247 0134.06401 CrossrefISIGoogle Scholar[52] Hisayoshi Shintani, Note on Miller's recurrence algorithm, J. Sci. Hiroshima Univ. Ser. A-I Math., 29 (1965), 121–133 MR0201048 0135.38703 CrossrefGoogle Scholar[53] Irene A. Stegun and , Milton Abramowitz, Generation of Coulomb wave functions by means of recurrence relations, Phys. Rev. (2), 98 (1955), 1851–1852 10.1103/PhysRev.98.1851 MR0070259 0065.43202 CrossrefISIGoogle Scholar[54] Irene A. Stegun and , Milton Abramowitz, Generation of Bessel functions on high speed computers, Math. Tables Aids Comput, 11 (1957), 255–257 MR0093939 0084.12101 CrossrefGoogle Scholar[55] Gabor Szegö, Orthogonal polynomials, American Mathematical Society Colloquium Publications, Vol. 23. Revised ed, American Mathematical Society, Providence, R.I., 1959ix+421 MR0106295 0089.27501 Google Scholar[56] D. Teichroew, Use of continued fractions in high speed computing, Math. Tables and Other Aids to Computation, 6 (1952), 127–133 MR0049650 0047.12002 CrossrefGoogle Scholar[57] E. Thieleker, Generating functions of confluent hypergeometric functions, unpublished Google Scholar[58] A. Tubis, Tables of nonrelativistic Coulomb wave functions, LA-2150, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, 1958 Google Scholar[59] H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc., New York, N. Y., 1948xiii+433 MR0025596 0035.03601 Google Scholar[60] H. Werner and , R. Collinge, Chebyshev approximations for the gamma function, Math. Comput., 15 (1961), 195–197 0099.11901 CrossrefGoogle Scholar[61] D. J. Wheeler, Personal communication, 1960 Google Scholar[62] Ion Zamfirescu, An extension of Gauss' method for the calculation of improper integrals, Acad. R. P. Romine Stud. Cerc. Mat., 14 (1963), 615–631 MR0184434 0235.65022 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Quasinormal modes and shadow of noncommutative black holeScientific Reports, Vol. 12, No. 1 | 20 May 2022 Cross Ref Hidden Spectral Symmetries and Mode Stability of Subextremal Kerr(-de Sitter) Black HolesCommunications in Mathematical Physics, Vol. 394, No. 2 | 5 July 2022 Cross Ref Glimpses of violation of strong cosmic censorship in rotating black holesPhysical Review D, Vol. 106, No. 4 | 26 August 2022 Cross Ref Computing Gaussian quadrature rules with high relative accuracyNumerical Algorithms, Vol. 18 | 2 June 2022 Cross Ref Nonlinear effects in the black hole ringdown: Absorption-induced mode excitationPhysical Review D, Vol. 105, No. 6 | 28 March 2022 Cross Ref Computation of solutions to linear difference and differential equations with a prescribed asymptotic behaviorAdvances in Difference Equations, Vol. 2021, No. 1 | 19 March 2021 Cross Ref Eigenvalue problems for a class of infinite complex symmetric tridiagonal matrices with related three-term recurrence relationJournal of Computational and Applied Mathematics, Vol. 194 | 1 Dec 2021 Cross Ref Orthogonal polynomials relative to a generalized Marchenko–Pastur probability measureNumerical Algorithms, Vol. 88, No. 3 | 18 March 2021 Cross Ref Quasinormal modes and shadow of a Schwarzschild black hole with GUPAnnals of Physics, Vol. 434 | 1 Nov 2021 Cross Ref Teukolsky master equation and Painlevé transcendents: Numerics and extremal limitPhysical Review D, Vol. 104, No. 8 | 13 October 2021 Cross Ref Resolving Singularities and Monodromy Reduction of Fuchsian ConnectionsAnnales Henri Poincaré, Vol. 22, No. 9 | 27 April 2021 Cross Ref Efficient numerical method for evaluating normal and anomalous time-domain equilibrium Green's functions in inhomogeneous systemsPhysical Review B, Vol. 104, No. 12 | 2 September 2021 Cross Ref Computation of Gabor coefficients for objects with polygonal cross section2021 International Conference on Electromagnetics in Advanced Applications (ICEAA) | 9 Aug 2021 Cross Ref VPA: Computer program for the computation of the phase shift in atom–atom potential scattering using the Variable Phase ApproachComputer Physics Communications, Vol. 263 | 1 Jun 2021 Cross Ref The aging effect in evolving scientific citation networksScientometrics, Vol. 126, No. 5 | 12 March 2021 Cross Ref Solutions of Heun's general equation and elliptic Darboux equationMathematical Methods in the Applied Sciences, Vol. 33 | 29 March 2021 Cross Ref Clenshaw–Curtis algorithms for an efficient numerical approximation of singular and highly oscillatory Fourier transform integralsJournal of Computational and Applied Mathematics, Vol. 385 | 1 Mar 2021 Cross Ref Alternative quantisation condition for wavepacket dynamics in a hyperbolic double wellJournal of Physics A: Mathematical and Theoretical, Vol. 54, No. 3 | 30 December 2020 Cross Ref Solving Poisson’s equation for birth–death chains: Structure, instability, and accurate approximationPerformance Evaluation, Vol. 145 | 1 Jan 2021 Cross Ref Additional methods for the stable calculation of isotropic Gaussian filter coefficients: The case of a truncated filter kernelComputers & Geosciences, Vol. 145 | 1 Dec 2020 Cross Ref Evolutionary dynamics and eigenspectrum of confluent Heun equationJournal of Physics A: Mathematical and Theoretical, Vol. 53, No. 39 | 28 August 2020 Cross Ref Analytic solutions of the Teukolsky equation for massless perturbations of any spin in de Sitter backgroundJournal of Mathematical Physics, Vol. 61, No. 10 | 1 Oct 2020 Cross Ref Encyclopedia of Special Functions: The Askey-Bateman Project14 September 2020 Cross Ref Excited hairy black holes: Dynamical construction and level transitionsPhysical Review D, Vol. 102, No. 4 | 6 August 2020 Cross Ref The Mott transition as a topological phase transitionPhysical Review B, Vol. 102, No. 8 | 14 August 2020 Cross Ref Matrix Calculus-Based Approach to Orthogonal Polynomial SequencesMediterranean Journal of Mathematics, Vol. 17, No. 4 | 25 June 2020 Cross Ref Quantum instability of the Cauchy horizon in Reissner–Nordström–deSitter spacetimeClassical and Quantum Gravity, Vol. 37, No. 11 | 19 May 2020 Cross Ref Noncommutative scalar field in the nonextremal Reissner-Nordström background: Quasinormal mode spectrumPhysical Review D, Vol. 101, No. 11 | 15 June 2020 Cross Ref Fast algorithms using orthogonal polynomialsActa Numerica, Vol. 29 | 30 November 2020 Cross Ref Quadrature rules from a RII type recurrence relation and associated quadrature rules on the unit circleNumerical Algorithms, Vol. 83, No. 3 | 3 May 2019 Cross Ref Complementary Romanovski–Routh Polynomials, Orthogonal Polynomials on the Unit Circle, and Extended Coulomb Wave FunctionsResults in Mathematics, Vol. 75, No. 1 | 13 February 2020 Cross Ref Classification of Laguerre–Hahn orthogonal polynomials of class oneMathematische Nachrichten, Vol. 293, No. 2 | 27 December 2019 Cross Ref Computing Hitting Probabilities of Markov Chains: Structural Results with regard to the Solution Space of the Corresponding System of EquationsJournal of Applied Mathematics, Vol. 2020 | 13 Jan 2020 Cross Ref The Mathematical Landscape up to the Mid-Twentieth CenturyExtrapolation and Rational Approximation | 1 December 2020 Cross Ref Constraint polynomial approach: an alternative to the functional Bethe Ansatz method?The European Physical Journal Plus, Vol. 135, No. 1 | 15 January 2020 Cross Ref The spectra of gravitational atomsJournal of Cosmology and Astroparticle Physics, Vol. 2019, No. 12 | 2 December 2019 Cross Ref Perturbations of an extremal Kerr spacetime: Analytic framework and late-time tailsPhysical Review D, Vol. 100, No. 12 | 10 December 2019 Cross Ref Generalized continued fractions: a unified definition and a Pringsheim-type convergence criterionAdvances in Difference Equations, Vol. 2019, No. 1 | 23 September 2019 Cross Ref Stable recurrent calculation of isotropic Gaussian filter coefficientsComputers & Geosciences, Vol. 133 | 1 Dec 2019 Cross Ref Contiguous relations, Laplace’s methods, and continued fractions for $${}_3F_2(1)$$ 3 F 2 ( 1 )The Ramanujan Journal, Vol. 49, No. 1 | 4 September 2018 Cross Ref Miscellaneous Topics on Symplectic SystemsSymplectic Difference Systems: Oscillation and Spectral Theory | 6 September 2019 Cross Ref Multilayered neural architectures evolution for computing sequences of orthogonal polynomialsAnnals of Mathematics and Artificial Intelligence, Vol. 84, No. 3-4 | 18 September 2018 Cross Ref Non-unitary evolution in the general extended EFT of inflation & excited initial statesJournal of High Energy Physics, Vol. 2018, No. 12 | 3 December 2018 Cross Ref Black Hole Quasibound States from a Draining Bathtub Vortex FlowPhysical Review Letters, Vol. 121, No. 6 | 7 August 2018 Cross Ref Laguerre-Freud equations for Generalized Hahn polynomials of type IJournal of Difference Equations and Applications, Vol. 24, No. 6 | 25 February 2018 Cross Ref Polynomial approximation for fast generation of associated Legendre functionsActa Geodaetica et Geophysica, Vol. 53, No. 2 | 28 April 2018 Cross Ref Initial value problem of space dynamics in universal Stumpff anomalyAstrophysics and Space Science, Vol. 363, No. 5 | 19 April 2018 Cross Ref Recursive Computation of Logarithmic Derivatives, Ratios, and Products of Spheroidal Harmonics and Modified Bessel Functions and ApplicationsJournal of Scientific Computing, Vol. 75, No. 1 | 5 August 2017 Cross Ref Subperiodic Trigonometric HyperinterpolationContemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan | 23 May 2018 Cross Ref Monotonic linear recurrencesLinear Algebra and its Applications, Vol. 537 | 1 Jan 2018 Cross Ref Stable fractional Chebyshev differentiation matrix for the numerical solution of multi-order fractional differential equationsNonlinear Dynamics, Vol. 90, No. 1 | 19 July 2017 Cross Ref Option pricing with Legendre polynomialsJournal of Computational and Applied Mathematics, Vol. 322 | 1 Oct 2017 Cross Ref On the 2-mode and k-photon quantum Rabi modelsReviews in Mathematical Physics, Vol. 29, No. 04 | 4 May 2017 Cross Ref A user-friendly method for computing indefinite integrals of oscillatory functionsJournal of Computational and Applied Mathematics, Vol. 315 | 1 May 2017 Cross Ref Steady-state distributions of ideal active Brownian particles under confinement and forcingJournal of Statistical Mechanics: Theory and Experiment, Vol. 2017, No. 4 | 5 April 2017 Cross Ref Poloidal and toroidal plasmons and fields of multilayer nanoringsPhysical Review B, Vol. 95, No. 16 | 17 April 2017 Cross Ref The quantization of the Rabi HamiltonianJournal of Physics A: Mathematical and Theoretical, Vol. 50, No. 11 | 14 February 2017 Cross Ref Numerical methods for the computation of the confluent and Gauss hypergeometric functionsNumerical Algorithms, Vol. 74, No. 3 | 27 August 2016 Cross Ref Analysis of the manufacturing signature using data miningPrecision Engineering, Vol. 47 | 1 Jan 2017 Cross Ref Inherent Numerical Instability in Computing Invariant Measures of Markov ChainsApplied Mathematics, Vol. 08, No. 09 | 1 Jan 2017 Cross Ref Purely imaginary quasinormal modes of the Kerr geometryClassical and Quantum Gravity, Vol. 33, No. 24 | 18 November 2016 Cross Ref A reduction formula for a q-beta integralAdvances in Difference Equations, Vol. 2016, No. 1 | 22 March 2016 Cross Ref Modes of the Kerr geometry with purely imaginary frequenciesPhysical Review D, Vol. 94, No. 10 | 30 November 2016 Cross Ref Convergence and applications of some solutions of the confluent Heun equationApplied Mathematics and Computation, Vol. 284 | 1 Jul 2016 Cross Ref Some Sets of Relative Stability Under Perturbations of Branched Continued Fractions with Complex Elements and a Variable Number of BranchesJournal of Mathematical Sciences, Vol. 215, No. 1 | 5 April 2016 Cross Ref Quasinormal modes of extremal black holesPhysical Review D, Vol. 93, No. 6 | 25 March 2016 Cross Ref Calculation of STOs electron repulsion integrals by ellipsoidal expansion and large-order approximationsJournal of Mathematical Chemistry, Vol. 54, No. 2 | 8 December 2015 Cross Ref Approximating Spectral Densities of Large MatricesLin Lin, Yousef Saad, and Chao YangSIAM Review, Vol. 58, No. 1 | 4 February 2016AbstractPDF (591 KB)Analytical Solutions of Basic Models in Quantum OpticsApplications + Practical Conceptualization + Mathematics = fruitful Innovation | 19 September 2015 Cross Ref Recurrences and Triangular SystemsParallelism in Matrix Computations | 26 July 2015 Cross Ref A fully spectral methodology for magnetohydrodynamic calculations in a whole sphereJournal of Computational Physics, Vol. 305 | 1 Jan 2016 Cross Ref Numerical Evaluation of 2D Ground StatesEPJ Web of Conferences, Vol. 108 | 9 February 2016 Cross Ref Electron-Proton Pairing at High Temperatures, Solar Flares, and the FIP EffectJournal of Modern Physics, Vol. 07, No. 01 | 1 Jan 2016 Cross Ref Efficient and accurate implementation of hp-BEM for the Laplace operator in 2DApplied Numerical Mathematics, Vol. 95 | 1 Sep 2015 Cross Ref Computational Analysis of SAXS Data AcquisitionJournal of Computational Biology, Vol. 22, No. 9 | 1 Sep 2015 Cross Ref The Kerr metricClassical and Quantum Gravity, Vol. 32, No. 12 | 1 June 2015 Cross Ref Integral relations for solutions of the confluent Heun equationApplied Mathematics and Computation, Vol. 256 | 1 Apr 2015 Cross Ref Calculation of the molecular integrals with the range-separated correlation factorThe Journal of Chemical Physics, Vol. 142, No. 12 | 28 Mar 2015 Cross Ref Gravitational waves from a particle in circular orbits around a rotating black hole to the 11th post-Newtonian orderProgress of Theoretical and Experimental Physics, Vol. 2015, No. 3 | 10 March 2015 Cross Ref Spectral functions and time evolution from the Chebyshev recursionPhysical Review B, Vol. 91, No. 11 | 31 March 2015 Cross Ref Recessive solutions for nonoscillatory discrete symplectic systemsLinear Algebra and its Applications, Vol. 469 | 1 Mar 2015 Cross Ref Special functions and spectrum of Jacobi matricesLinear Algebra and its Applications, Vol. 464 | 1 Jan 2015 Cross Ref Numerical computation of the critical energy constant for two-dimensional Boussinesq equations1 Jan 2015 Cross Ref Continued Fraction Evaluation of the Universal Y’s FunctionsInternational Journal of Astronomy and Astrophysics, Vol. 05, No. 01 | 1 Jan 2015 Cross Ref Quantum models with spectrum generated by the flows of polynomial zerosJournal of Physics A: Mathematical and Theoretical, Vol. 47, No. 49 | 20 November 2014 Cross Ref Haydock’s recursive solution of self-adjoint problems. Discrete spectrumAnnals of Physics, Vol. 351 | 1 Dec 2014 Cross Ref Chandrasekhar Polynomials and the Solution to the Transport Equation in an Infinite MediumJournal of Computational and Theoretical Transport, Vol. 43, No. 1-7 | 22 December 2014 Cross Ref Quasinormal modes of charged fields around a Reissner-Nordström black holePhysical Review D, Vol. 90, No. 12 | 3 December 2014 Cross Ref Gravitational perturbations of the Kerr geometry: High-accuracy studyPhysical Review D, Vol. 90, No. 12 | 5 December 2014 Cross Ref Reexamination of the calculation of two-center, two-electron integrals over Slater-type orbitals. I. Coulomb and hybrid integralsPhysical Review E, Vol. 90, No. 6 | 29 December 2014 Cross Ref Reexamination of the calculation of two-center, two-electron integrals over Slater-type orbitals. II. Neumann expansion of the exchange integralsPhysical Review E, Vol. 90, No. 6 | 29 December 2014 Cross Ref Recent software developments for special functions in the Santander–Amsterdam projectScience of Computer Programming, Vol. 90 | 1 Sep 2014 Cross Ref ZN symmetric chiral Rabi model: A new N -level systemAnnals of Physics, Vol. 347 | 1 Aug 2014 Cross Ref ON MULTISERVER RETRIAL QUEUES: HISTORY, OKUBO-TYPE HYPERGEOMETRIC SYSTEMS AND MATRIX CONTINUED-FRACTIONSAsia-Pacific Journal of Operational Research, Vol. 31, No. 02 | 9 May 2014 Cross Ref Mass hierarchy, mass gap and corrections to Newton’s law on thick branes with Poincaré symmetryGeneral Relativity and Gravitation, Vol. 46, No. 1 | 19 December 2013 Cross Ref Asymptotic Expansions of Barnett–Coulson–Löwdin Functions of High OrderProceedings of MEST 2012: Electronic structure methods with applications to experimental chemistry | 1 Jan 2014 Cross Ref A hidden analytic structure of the Rabi modelAnnals of Physics, Vol. 340, No. 1 | 1 Jan 2014 Cross Ref On Fast and Stable Implementation of Clenshaw-Curtis and Fejér-Type Quadrature RulesAbstract and Applied Analysis, Vol. 2014 | 1 Jan 2014 Cross Ref Solving the two-mode squeezed harmonic oscillator and the k th-order harmonic generation in Bargmann–Hilbert spacesJournal of Physics A: Mathematical and Theoretical, Vol. 46, No. 45 | 24 October 2013 Cross Ref On solvability and integrability of the Rabi modelAnnals of Physics, Vol. 338 | 1 Nov 2013 Cross Ref Hill’s determinant approach to single-mode spin-boson modelJournal of Physics A: Mathematical and Theoretical, Vol. 46, No. 43 | 11 October 2013 Cross Ref Stability of rapidly rotating charged black holes in AdS 5 × S 5Classical and Quantum Gravity, Vol. 30, No. 12 | 16 May 2013 Cross Ref The characteristic function for Jacobi matrices with applicationsLinear Algebra and its Applications, Vol. 438, No. 11 | 1 Jun 2013 Cross Ref Continued fractions and the Rabi modelJournal of Physics A: Mathematical and Theoretical, Vol. 46, No. 17 | 9 April 2013 Cross Ref Computation of integrals with oscillatory and singular integrands using Chebyshev expansionsJournal of Computational and Applied Mathematics, Vol. 242 | 1 Apr 2013 Cross Ref A generalized G -function for the Quantum Rabi ModelAnnalen der Physik, Vol. 525, No. 3 | 14 February 2013 Cross Ref A Pringsheim-type convergence criterion for continued fractions in Banach algebrasJournal of Approximation Theory, Vol. 166 | 1 Feb 2013 Cross Ref On the spectrum of a class of quantum modelsEPL (Europhysics Letters), Vol. 100, No. 6 | 7 January 2013 Cross Ref Degenerate rotating black holes, chiral CFTs and Fermi surfaces I — Analytic results for quasinormal modesJournal of High Energy Physics, Vol. 2012, No. 8 | 22 August 2012 Cross Ref Exact evaluation of the quadratic longitudinal response function for an unmagnetized Maxwellian plasmaPhysics of Plasmas, Vol. 19, No. 7 | 1 Jul 2012 Cross Ref High-precision computation: Mathematical physics and dynamicsApplied Mathematics and Computation, Vol. 218, No. 20 | 1 Jun 2012 Cross Ref An Efficient Algorithm for the Computation of the UTD $T$ Transition FunctionIEEE Transactions on Antennas and Propagation, Vol. 60, No. 5 | 1 May 2012 Cross Ref Resolutions of the Coulomb operator. VI. Computation of auxiliary integralsThe Journal of Chemical Physics, Vol. 136, No. 10 | 14 Mar 2012 Cross Ref Numerical calculation of Bessel, Hankel and Airy functionsComputer Physics Communications, Vol. 183, No. 3 | 1 Mar 2012 Cross Ref Realistic Analytical Phantoms for Parallel Magnetic Resonance ImagingIEEE Transactions on Medical Imaging, Vol. 31, No. 3 | 1 Mar 2012 Cross Ref Details of the Envelope FunctionsThe Augmented Spherical Wave Method | 1 Jan 2012 Cross Ref Supersymmetric Quantum Mechanics with Lévy Disorder in One DimensionJournal of Statistical Physics, Vol. 145, No. 5 | 24 September 2011 Cross Ref Validated computation of certain hypergeometric functionsACM Transactions on Mathematical Software, Vol. 38, No. 2 | 1 Dec 2011 Cross Ref Algorithm for the Computation of the Generalized Fresnel IntegralIEEE Transactions on Antennas and Propagation, Vol. 59, No. 10 | 1 Oct 2011 Cross Ref À la carte recurrence relations for continuous and discrete hypergeometric functionsSeMA Journal, Vol. 55, No. 1 | 19 April 2013 Cross Ref Orthogonal polynomial expansions for the matrix exponentialLinear Algebra and its Applications, Vol. 435, No. 3 | 1 Aug 2011 Cross Ref Scaling dimensions in hidden Kerr/CFT correspondencePhysical Review D, Vol. 84, No. 2 | 18 July 2011 Cross Ref BibliographySpecial Functions | 1 March 2011 Cross Ref Basic Methods for Computing Special FunctionsRecent Advances in Computational and Appli