Computational Aspects of Three-Term Recurrence Relations

Previous article Next article Computational Aspects of Three-Term Recurrence RelationsWalter GautschiWalter Gautschihttps: //doi. org/10. 1137/1009002PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout1 M. Abramowitz, Review 58, Math. Tables Aids Comput, 10 (1956), 176– Google Scholar2 Renata Babušková, Über numerische Stabilität einiger Rekursionsformeln, Apl. Mat. , 9 (1964), 186–193 MR0187365 0123. 11104 Google Scholar3 T. A. Bancroft, Some recurrence formulae in the incomplete beta function ratio, Ann. Math. Statistics, 20 (1949), 451–455 MR0030653 0041. 03402 CrossrefISIGoogle Scholar4 G. Blanch, Numerical evaluation of continued fractions, SIAM Rev. , 6 (1964), 383–421 10. 1137/1006092 MR0171374 0133. 38904 LinkISIGoogle Scholar5 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 Scholar6 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 Scholar7 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 Scholar8 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 Scholar9 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 Scholar10 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 Scholar11 A. Erdélyi, Asymptotic expansions, Dover Publications Inc. , New York, 1956vi+108 MR0078494 0070. 29002 Google Scholar12 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 Scholar13 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 Scholar14 M. A. Efgrafov, A new proof of Perron's theorem, Izvestiya Akad. Nauk SSSR. Ser. Mat. , 17 (1953), 77–82 MR0058109 Google Scholar15 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 Scholar16 Walter Gautschi, Recursive computation of the repeated integrals of the error function, Math. Comp. , 15 (1961), 227–232 MR0136074 0099. 11401 Google Scholar17 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 Scholar18 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 Scholar19 W. Gautschi, Algorithm 259—Legendre functions for arguments larger than one, Comm. ACM, 8 (1965), 488–492 10. 1145/365474. 355591 CrossrefISIGoogle Scholar20 W. Gautschi, Algorithm 292—Regular Coulomb wave functions, Comm. ACM, 9 (1966), 793–795 10. 1145/365876. 365884 CrossrefISIGoogle Scholar21 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 Scholar22 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 Scholar23 Marion C. Gray, Bessel functions of integral order and complex argument, Comm. ACM, 4 (1961), 169– 10. 1145/355578. 366318 MR0151652 CrossrefISIGoogle Scholar24 Philip Hartman, Ordinary differential equations, John Wiley & Sons Inc. , New York, 1964xiv+612 MR0171038 0125. 32102 Google Scholar25 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 Scholar26 Sin Hitotumatu, Note on the computation of Bessel functions through recurrence formula, J. Math. Soc. Japan, 15 (1963), 353–359 MR0156008 0131. 34401 CrossrefGoogle Scholar27 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 Scholar28 W. Kahan, Note on bounds for generating Bessel functions by recurrence, un published Google Scholar29 P. Kreuser, Masters Thesis, Über das Verhalten der Integrale homogener linearer Differenzengleichungen im Unendlichen, Dissertation, University of Tübingen, Leipzig, 1914 Google Scholar30 N. A. Logan, Survey of some early studies of the scattering of plane waves by a sphere, Proc. IEEE, 53 (1965), 773–785 CrossrefISIGoogle Scholar31 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 Scholar32 Saburo Makinouchi, Note on the recurrence techniques for the calculation of Bessel functions Jv (x), Tech. Rep. Osaka Univ. , 16 (1965), 185–201 MR0198658 Google Scholar33 Saburo Makinouchi, Note on the recurrence techniques for the calculation of Bessel functions Iv (x), Tech. Rep. Osaka Univ. , 16 (1965), 203–216 MR0198659 Google Scholar34 Herbert Meschkowski, Differenzengleichungen, Studia Mathematica, Bd. XIV, Vandenhoeck & Ruprecht, Göttingen, 1959, 243– MR0109264 0084. 29202 Google Scholar35 L. M. Milne-Thomson, The Calculus of Finite Differences, Macmillan, London, 1933 0008. 01801 Google Scholar36 Google Scholar37 N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer, Berlin, 1924 CrossrefGoogle Scholar38 F. W. J. Olver, Error analysis of Miller's recurrence algorithm, Math. Comp. , 18 (1964), 65–74 MR0169406 0115. 34502 ISIGoogle Scholar39 F. W. J. Olver, Bessel functions of integer order, Handbook of Mathematical Functions, NBS Appl. Math. Ser. 55, 1964, Chap. 9 Google Scholar40 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 Scholar41 O. Perron, Über einen Satz des Herrn Poincaré, J. Reine Angew. Math. , 136 (1909), 17–37 CrossrefGoogle Scholar42 O. Perron, Über lineare Differenzengleichungen, Acta Math. , 34 (1911), 109–137 CrossrefGoogle Scholar43A Oskar Perron, Die Lehre von den Kettenbrüchen. Bd I. Elementare Kettenbrüche, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1954vi+194 MR0064172 0056. 05901 Google Scholar43B 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 Scholar44 S. Pincherle, Sur la génération de systèmes récurrents au moyen d'une équation linéaire différentielle, Acta Math. , 16 (1892), 341–363 CrossrefGoogle Scholar45 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 Scholar46 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 Scholar47 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 Scholar48 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 Scholar49 Louis Robin, Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome II, Gauthier-Villars, Paris, 1958viii+384 MR0101928 0088. 05102 Google Scholar50 A. Rotenberg, The calculation of toroidal harmonics, Math. Comput. , 14 (1960), 274–276 MR0115264 0096. 32205 CrossrefGoogle Scholar51 F. W. Schäfke, Lösungstypen von Differenzengleichungen und Summengleichungen in normierten abelschen Gruppen, Math. Z. , 88 (1965), 61–104 MR0176247 0134. 06401 CrossrefISIGoogle Scholar52 Hisayoshi Shintani, Note on Miller's recurrence algorithm, J. Sci. Hiroshima Univ. Ser. A-I Math. , 29 (1965), 121–133 MR0201048 0135. 38703 CrossrefGoogle Scholar53 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 Scholar54 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 Scholar55 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 Scholar56 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 Scholar57 E. Thieleker, Generating functions of confluent hypergeometric functions, unpublished Google Scholar58 A. Tubis, Tables of nonrelativistic Coulomb wave functions, LA-2150, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, 1958 Google Scholar59 H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc. , New York, N. Y. , 1948xiii+433 MR0025596 0035. 03601 Google Scholar60 H. Werner and, R. Collinge, Chebyshev approximations for the gamma function, Math. Comput. , 15 (1961), 195–197 0099. 11901 CrossrefGoogle Scholar61 D. J. Wheeler, Personal communication, 1960 Google Scholar62 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 ₃F₂ (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. | 18 September 2018 Cross Ref evolution in the general of & of Physics, Vol. No. 12 | 3 December 2018 Cross Ref Black from a Review Vol. No. 6 | 7 August 2018 Cross Ref equations for Generalized polynomials of type of Difference Equations and Vol. No. 6 | 25 February 2018 Cross Ref Polynomial approximation for of associated Legendre et Vol. 53, No. 2 | 28 April 2018 Cross Ref of dynamics in and Space Science, Vol. No. 5 | 19 April 2018 Cross Ref Recursive Computation of and of and Bessel Functions and of Scientific Vol. 75, No. 1 | 5 August Cross Ref Computational Mathematics A of the of | 23 May 2018 Cross Ref linear and Vol. | 1 Jan 2018 Cross Ref Stable Chebyshev for the numerical solution of differential Vol. No. 1 | 19 July Cross Ref with Legendre of Computational and Applied Mathematics, Vol. | 1 Oct Cross Ref On the and in Mathematical Physics, Vol. No. | 4 May Cross Ref A method for computing integrals of oscillatory of Computational and Applied Mathematics, Vol. | 1 May Cross Ref of and of Theory and Vol. No. 4 | 5 April Cross Ref and toroidal and of Review B, Vol. No. 16 | 17 April Cross Ref The of the of Physics A: Mathematical and Theoretical, Vol. No. 11 | 14 February Cross Ref Numerical methods for the computation of the confluent and hypergeometric Algorithms, Vol. No. 3 | 27 August Cross Ref of the using Vol. | 1 Jan Cross Ref Numerical in Computing of Markov Mathematics, Vol. No. | 1 Jan Cross Ref imaginary modes of the Kerr and Quantum Gravity, Vol. No. 24 | 18 November Cross Ref A for a in Difference Equations, Vol. No. 1 | March Cross Ref of the Kerr with imaginary Review D, Vol. No. 10 | 30 November Cross Ref and applications of some solutions of the confluent Heun Mathematics and Computation, Vol. | 1 Cross Ref Some of Stability Perturbations of Continued with and a Variable of of Mathematical Sciences, Vol. No. 1 | 5 April Cross Ref Quasinormal modes of extremal black holesPhysical Review D, Vol. No. 6 | 25 March Cross Ref of integrals by and of Mathematical Vol. 54, No. 2 | 8 December Cross Ref Spectral of and Vol. 58, No. 1 | 4 February Solutions of in Quantum Mathematics | 19 September Cross Ref and in Matrix | 26 July Cross Ref A for in a of Computational Physics, Vol. | 1 Jan Cross Ref Numerical of of Vol. | 9 February Cross Ref at and the of Physics, Vol. No. | 1 Jan Cross Ref Efficient and accurate of for the in Numerical Mathematics, Vol. | 1 Cross Ref Computational of of Computational Vol. 22, No. 9 | 1 Cross Ref The Kerr and Quantum Gravity, Vol. No. 12 | 1 June Cross Ref for solutions of the confluent Heun Mathematics and Computation, Vol. | 1 Cross Ref of the integrals with the Journal of Physics, Vol. No. 12 | 28 Mar Cross Ref waves from a in a rotating black hole to the of and Physics, Vol. No. 3 | 10 March Cross Ref Spectral functions and evolution from the Chebyshev Review B, Vol. No. 11 | March Cross Ref solutions for and Vol. | 1 Mar Cross Ref Special functions and of and Vol. | 1 Jan Cross Ref Numerical computation of the for Jan Cross Ref Continued of the Journal of and Vol. No. | 1 Jan Cross Ref Quantum with by the of polynomial of Physics A: Mathematical and Theoretical, Vol. No. | 20 November Cross Ref solution of of Physics, Vol. | 1 Dec Cross Ref Polynomials and the Solution to the in an of Computational and Vol. No. | December Cross Ref Quasinormal modes of a Reissner-Nordström black Review D, Vol. No. 12 | 3 December Cross Ref perturbations of the Kerr Review D, Vol. No. 12 | 5 December Cross Ref of the calculation of integrals I. Coulomb and Review Vol. No. 6 | 29 December Cross Ref of the calculation of integrals II. of the Review Vol. No. 6 | 29 December Cross Ref for functions in the of Computer Vol. | 1 Cross Ref symmetric A new of Physics, Vol. | 1 Aug Cross Ref Journal of Vol. No. | 9 May Cross Ref and to on with and Vol. No. 1 | 19 December Cross Ref Asymptotic of Functions of of methods with applications to | 1 Jan Cross Ref A of the of Physics, Vol. No. 1 | 1 Jan Cross Ref On Fast and Stable of and Quadrature and Applied Vol. | 1 Jan Cross Ref Solving the and the in of Physics A: Mathematical and Theoretical, Vol. No. | 24 October Cross Ref On and of the of Physics, Vol. | 1 Nov Cross Ref to of Physics A: Mathematical and Theoretical, Vol. No. | 11 October Cross Ref Stability of rotating black in 5 and Quantum Gravity, Vol. No. 12 | 16 May Cross Ref The function for matrices with and Vol. No. 11 | 1 Jun Cross Ref Continued fractions and the of Physics A: Mathematical and Theoretical, Vol. No. 17 | 9 April Cross Ref Computation of integrals with oscillatory and singular using Chebyshev of Computational and Applied Mathematics, Vol. | 1 Cross Ref A generalized for the Quantum der Vol. No. 3 | 14 February Cross Ref A Pringsheim-type convergence for continued fractions in of Approximation Vol. | 1 Cross Ref On the of a class of Vol. 100, No. 6 | 7 January Cross Ref rotating black and Analytic for of Physics, Vol. No. 8 | August Cross Ref evaluation of the function for an of Vol. No. 7 | 1 Cross Ref Mathematical and Mathematics and Computation, Vol. No. 20 | 1 Jun Cross Ref An Efficient Algorithm for the Computation of the on and Vol. 60, No. 5 | 1 May Cross Ref of the Coulomb Computation of auxiliary Journal of Physics, Vol. No. 10 | 14 Mar Cross Ref Numerical calculation of and Physics Communications, Vol. No. 3 | 1 Mar Cross Ref for on Vol. No. 3 | 1 Mar Cross Ref of the Wave | 1 Jan Cross Ref Quantum with in of Physics, Vol. No. 5 | 24 September Cross Ref computation of hypergeometric on Mathematical Vol. No. 2 | 1 Dec Cross Ref Algorithm for the Computation of the Generalized on and Vol. No. 10 | 1 Oct Cross Ref la recurrence for and hypergeometric Journal, Vol. 55, No. 1 | 19 April Cross Ref Orthogonal polynomial for the and Vol. No. 3 | 1 Aug Cross Ref dimensions in Review D, Vol. 84, No. 2 | 18 July Cross Ref Functions | 1 March Cross Ref Methods for Computing Special in Computational and Applied