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11: Bibliography R
  • S. Ramanujan (1927) Some properties of Bernoulli’s numbers (J. Indian Math. Soc. 3 (1911), 219–234.). In Collected Papers,
  • W. H. Reid (1974b) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory. Studies in Appl. Math. 53, pp. 217–224.
  • W. P. Reinhardt (2021a) Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (4), pp. 91.
  • W. P. Reinhardt (2021b) Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (3), pp. 56–64.
  • R. R. Rosales (1978) The similarity solution for the Korteweg-de Vries equation and the related Painlevé transcendent. Proc. Roy. Soc. London Ser. A 361, pp. 265–275.
  • 12: Bibliography I
  • K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
  • A. Iserles, P. E. Koch, S. P. Nørsett, and J. M. Sanz-Serna (1991) On polynomials orthogonal with respect to certain Sobolev inner products. J. Approx. Theory 65 (2), pp. 151–175.
  • M. E. H. Ismail, D. R. Masson, and M. Rahman (Eds.) (1997) Special Functions, q -Series and Related Topics. Fields Institute Communications, Vol. 14, American Mathematical Society, Providence, RI.
  • M. E. H. Ismail and D. R. Masson (1991) Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.
  • M. E. H. Ismail (2000a) An electrostatics model for zeros of general orthogonal polynomials. Pacific J. Math. 193 (2), pp. 355–369.
  • 13: Bibliography W
  • S. S. Wagstaff (2002) Prime Divisors of the Bernoulli and Euler Numbers. In Number Theory for the Millennium, III (Urbana, IL, 2000), pp. 357–374.
  • X.-S. Wang and R. Wong (2012) Asymptotics of orthogonal polynomials via recurrence relations. Anal. Appl. (Singap.) 10 (2), pp. 215–235.
  • J. A. Wilson (1978) Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.
  • J. Wimp (1984) Computation with Recurrence Relations. Pitman, Boston, MA.
  • R. Wong and J.-M. Zhang (1997) Asymptotic expansions of the generalized Bessel polynomials. J. Comput. Appl. Math. 85 (1), pp. 87–112.
  • 14: 24.17 Mathematical Applications
    §24.17 Mathematical Applications
    Bernoulli Monosplines
    §24.17(iii) Number Theory
    Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and L -series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); p -adic analysis (Koblitz (1984, Chapter 2)).
    15: Bibliography K
  • M. Kaneko (1997) Poly-Bernoulli numbers. J. Théor. Nombres Bordeaux 9 (1), pp. 221–228.
  • T. Kasuga and R. Sakai (2003) Orthonormal polynomials with generalized Freud-type weights. J. Approx. Theory 121 (1), pp. 13–53.
  • N. Kimura (1988) On the degree of an irreducible factor of the Bernoulli polynomials. Acta Arith. 50 (3), pp. 243–249.
  • T. H. Koornwinder (2007b) The structure relation for Askey-Wilson polynomials. J. Comput. Appl. Math. 207 (2), pp. 214–226.
  • T. Kriecherbauer and K. T.-R. McLaughlin (1999) Strong asymptotics of polynomials orthogonal with respect to Freud weights. Internat. Math. Res. Notices 1999 (6), pp. 299–333.
  • 16: Bibliography S
  • D. Schmidt and G. Wolf (1979) A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM J. Math. Anal. 10 (4), pp. 823–838.
  • I. Sh. Slavutskiĭ (2000) On the generalized Bernoulli numbers that belong to unequal characters. Rev. Mat. Iberoamericana 16 (3), pp. 459–475.
  • S. Yu. Slavyanov and N. A. Veshev (1997) Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.
  • D. Slepian (1964) Prolate spheroidal wave functions, Fourier analysis and uncertainity. IV. Extensions to many dimensions; generalized prolate spheroidal functions. Bell System Tech. J. 43, pp. 3009–3057.
  • F. C. Smith (1939b) Relations among the fundamental solutions of the generalized hypergeometric equation when p = q + 1 . II. Logarithmic cases. Bull. Amer. Math. Soc. 45 (12), pp. 927–935.
  • 17: Bibliography D
  • K. Dilcher (1987a) Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials. J. Approx. Theory 49 (4), pp. 321–330.
  • K. Dilcher (1987b) Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters. J. Number Theory 25 (1), pp. 72–80.
  • K. Dilcher (1988) Zeros of Bernoulli, generalized Bernoulli and Euler polynomials. Mem. Amer. Math. Soc. 73 (386), pp. iv+94.
  • K. Dilcher (2008) On multiple zeros of Bernoulli polynomials. Acta Arith. 134 (2), pp. 149–155.
  • T. M. Dunster (2001c) Uniform asymptotic expansions for the reverse generalized Bessel polynomials, and related functions. SIAM J. Math. Anal. 32 (5), pp. 987–1013.
  • 18: Bibliography G
  • W. Gautschi (1959b) Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.
  • W. Gautschi (1984) Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.
  • H. W. Gould (1972) Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.
  • W. Groenevelt (2007) Fourier transforms related to a root system of rank 1. Transform. Groups 12 (1), pp. 77–116.
  • X. Guan, O. Zatsarinny, K. Bartschat, B. I. Schneider, J. Feist, and C. J. Noble (2007) General approach to few-cycle intense laser interactions with complex atoms. Phys. Rev. A 76, pp. 053411.
  • 19: Errata
    The specific updates to Chapter 18 include some results for general orthogonal polynomials including quadratic transformations, uniqueness of orthogonality measure and completeness, moments, continued fractions, and some special classes of orthogonal polynomials. …
  • Chapter 18 Additions

    The following additions were made in Chapter 18:

    • Section 18.2

      In Subsection 18.2(i), Equation (18.2.1_5); the paragraph title “Orthogonality on Finite Point Sets” has been changed to “Orthogonality on Countable Sets”, and there are minor changes in the presentation of the final paragraph, including a new equation (18.2.4_5). The presentation of Subsection 18.2(iii) has changed, Equation (18.2.5_5) was added and an extra paragraph on standardizations has been included. The presentation of Subsection 18.2(iv) has changed and it has been expanded with two extra paragraphs and several new equations, (18.2.9_5), (18.2.11_1)–(18.2.11_9). Subsections 18.2(v) (with (18.2.12_5), (18.2.14)–(18.2.17)) and 18.2(vi) (with (18.2.17)–(18.2.20)) have been expanded. New subsections, 18.2(vii)18.2(xii), with Equations (18.2.21)–(18.2.46),

    • Section 18.3

      A new introduction, minor changes in the presentation, and three new paragraphs.

    • Section 18.5

      Extra details for Chebyshev polynomials, and Equations (18.5.4_5), (18.5.11_1)–(18.5.11_4), (18.5.17_5).

    • Section 18.8

      Line numbers and two extra rows were added to Table 18.8.1.

    • Section 18.9

      Subsection 18.9(i) has been expanded, and 18.9(iii) has some additional explanation. Equations (18.9.2_1), (18.9.2_2), (18.9.18_5) and Table 18.9.2 were added.

    • Section 18.12

      Three extra generating functions, (18.12.2_5), (18.12.3_5), (18.12.17).

    • Section 18.14

      Equation (18.14.3_5). New subsection, 18.14(iv), with Equations (18.14.25)–(18.14.27).

    • Section 18.15

      Equation (18.15.4_5).

    • Section 18.16

      The title of Subsection 18.16(iii) was changed from “Ultraspherical and Legendre” to “Ultraspherical, Legendre and Chebyshev”. New subsection, 18.16(vii) Discriminants, with Equations (18.16.19)–(18.16.21).

    • Section 18.17

      Extra explanatory text at many places and seven extra integrals (18.17.16_5), (18.17.21_1)–(18.17.21_3), (18.17.28_5), (18.17.34_5), (18.17.41_5).

    • Section 18.18

      Extra explanatory text at several places and the title of Subsection 18.18(iv) was changed from “Connection Formulas” to “Connection and Inversion Formulas”.

    • Section 18.19

      A new introduction.

    • Section 18.21

      Equation (18.21.13).

    • Section 18.25

      Extra explanatory text in Subsection 18.25(i) and the title of Subsection 18.25(ii) was changed from “Weights and Normalizations: Continuous Cases” to “Weights and Standardizations: Continuous Cases”.

    • Section 18.26

      In Subsection 18.26(i) an extra paragraph on dualities has been included, with Equations (18.26.4_1), (18.26.4_2).

    • Section 18.27

      Extra text at the start of this section and twenty seven extra formulas, (18.27.4_1), (18.27.4_2), (18.27.6_5), (18.27.9_5), (18.27.12_5), (18.27.14_1)–(18.27.14_6), (18.27.17_1)–(18.27.17_3), (18.27.20_5), (18.27.25), (18.27.26), (18.28.1_5).

    • Section 18.28

      A big expansion. Six extra formulas in Subsection 18.28(ii) ((18.28.6_1)–(18.28.6_5)) and three extra formulas in Subsection 18.28(viii) ((18.28.21)–(18.28.23)). New subsections, 18.28(ix)18.28(xi), with Equations (18.28.23)–(18.28.34).

    • Section 18.30

      Originally this section did not have subsections. The original seven formulas have now more explanatory text and are split over two subsections. New subsections 18.30(iii)18.30(viii), with Equations (18.30.8)–(18.30.31).

    • Section 18.32

      This short section has been expanded, with Equation (18.32.2).

    • Section 18.33

      Additional references and a new large subsection, 18.33(vi), including Equations (18.33.17)–(18.33.32).

    • Section 18.34

      This section has been expanded, including an extra orthogonality relations (18.34.5_5), (18.34.7_1)–(18.34.7_3).

    • Section 18.35

      This section on Pollaczek polynomials has been significantly updated with much more explanations and as well to include the Pollaczek polynomials of type 3 which are the most general with three free parameters. The Pollaczek polynomials which were previously treated, namely those of type 1 and type 2 are special cases of the type 3 Pollaczek polynomials. In the first paragraph of this section an extensive description of the relations between the three types of Pollaczek polynomials is given which was lacking previously. Equations (18.35.0_5), (18.35.2_1)–(18.35.2_5), (18.35.4_5), (18.35.6_1)–(18.35.6_6), (18.35.10).

    • Section 18.36

      This section on miscellaneous polynomials has been expanded with new subsections, 18.36(v) on non-classical Laguerre polynomials and 18.36(vi) with examples of exceptional orthogonal polynomials, with Equations (18.36.1)–(18.36.10). In the titles of Subsections 18.36(ii) and 18.36(iii) we replaced “OP’s” by “Orthogonal Polynomials”.

    • Section 18.38

      The paragraphs of Subsection 18.38(i) have been re-ordered and one paragraph was added. The title of Subsection 18.38(ii) was changed from “Classical OP’s: Other Applications” to “Classical OP’s: Mathematical Developments and Applications”. Subsection 18.38(iii) has been expanded with seven new paragraphs, and Equations (18.38.4)–(18.38.11).

    • Section 18.39

      This section was completely rewritten. The previous 18.39(i) Quantum Mechanics has been replaced by Subsections 18.39(i) Quantum Mechanics and 18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom, containing the same essential information; the original content of the subsection is reproduced below for reference. Subsection 18.39(ii) was moved to 18.39(v) Other Applications. New subsections, 18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences, 18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods; Equations (18.39.7)–(18.39.48); and Figures 18.39.1, 18.39.2.

      The original text of 18.39(i) Quantum Mechanics was:

      “Classical OP’s appear when the time-dependent Schrödinger equation is solved by separation of variables. Consider, for example, the one-dimensional form of this equation for a particle of mass m with potential energy V ( x ) :

      errata.1 ( 2 2 m 2 x 2 + V ( x ) ) ψ ( x , t ) = i t ψ ( x , t ) ,

      where is the reduced Planck’s constant. On substituting ψ ( x , t ) = η ( x ) ζ ( t ) , we obtain two ordinary differential equations, each of which involve the same constant E . The equation for η ( x ) is

      errata.2 d 2 η d x 2 + 2 m 2 ( E V ( x ) ) η = 0 .

      For a harmonic oscillator, the potential energy is given by

      errata.3 V ( x ) = 1 2 m ω 2 x 2 ,

      where ω is the angular frequency. For (18.39.2) to have a nontrivial bounded solution in the interval < x < , the constant E (the total energy of the particle) must satisfy

      errata.4 E = E n = ( n + 1 2 ) ω , n = 0 , 1 , 2 , .

      The corresponding eigenfunctions are

      errata.5 η n ( x ) = π 1 4 2 1 2 n ( n ! b ) 1 2 H n ( x / b ) e x 2 / 2 b 2 ,

      where b = ( / m ω ) 1 / 2 , and H n is the Hermite polynomial. For further details, see Seaborn (1991, p. 224) or Nikiforov and Uvarov (1988, pp. 71-72).

      A second example is provided by the three-dimensional time-independent Schrödinger equation

      errata.6 2 ψ + 2 m 2 ( E V ( 𝐱 ) ) ψ = 0 ,

      when this is solved by separation of variables in spherical coordinates (§1.5(ii)). The eigenfunctions of one of the separated ordinary differential equations are Legendre polynomials. See Seaborn (1991, pp. 69-75).

      For a third example, one in which the eigenfunctions are Laguerre polynomials, see Seaborn (1991, pp. 87-93) and Nikiforov and Uvarov (1988, pp. 76-80 and 320-323).”

    • Section 18.40

      The old section is now Subsection 18.40(i) and a large new subsection, 18.40(ii), on the classical moment problem has been added, with formulae (18.40.1)–(18.40.10) and Figures 18.40.1, 18.40.2.

  • Chapters 14 Legendre and Related Functions, 15 Hypergeometric Function

    The Gegenbauer function C α ( λ ) ( z ) , was labeled inadvertently as the ultraspherical (Gegenbauer) polynomial C n ( λ ) ( z ) . In order to resolve this inconsistency, this function now links correctly to its definition. This change affects Gegenbauer functions which appear in §§14.3(iv), 15.9(iii).

  • Equations (25.11.6), (25.11.19), and (25.11.20)

    Originally all six integrands in these equations were incorrect because their numerators contained the function B ~ 2 ( x ) . The correct function is B ~ 2 ( x ) B 2 2 . The new equations are:

    25.11.6 ζ ( s , a ) = 1 a s ( 1 2 + a s 1 ) s ( s + 1 ) 2 0 B ~ 2 ( x ) B 2 ( x + a ) s + 2 d x , s 1 , s > 1 , a > 0

    Reported 2016-05-08 by Clemens Heuberger.

    25.11.19 ζ ( s , a ) = ln a a s ( 1 2 + a s 1 ) a 1 s ( s 1 ) 2 + s ( s + 1 ) 2 0 ( B ~ 2 ( x ) B 2 ) ln ( x + a ) ( x + a ) s + 2 d x ( 2 s + 1 ) 2 0 B ~ 2 ( x ) B 2 ( x + a ) s + 2 d x , s > 1 , s 1 , a > 0

    Reported 2016-06-27 by Gergő Nemes.

    25.11.20 ( 1 ) k ζ ( k ) ( s , a ) = ( ln a ) k a s ( 1 2 + a s 1 ) + k ! a 1 s r = 0 k 1 ( ln a ) r r ! ( s 1 ) k r + 1 s ( s + 1 ) 2 0 ( B ~ 2 ( x ) B 2 ) ( ln ( x + a ) ) k ( x + a ) s + 2 d x + k ( 2 s + 1 ) 2 0 ( B ~ 2 ( x ) B 2 ) ( ln ( x + a ) ) k 1 ( x + a ) s + 2 d x k ( k 1 ) 2 0 ( B ~ 2 ( x ) B 2 ) ( ln ( x + a ) ) k 2 ( x + a ) s + 2 d x , s > 1 , s 1 , a > 0

    Reported 2016-06-27 by Gergő Nemes.

  • Equation (10.13.4)

    has been generalized to cover an additional case.

  • 20: Bibliography N
  • National Bureau of Standards (1967) Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..
  • J. Negro, L. M. Nieto, and O. Rosas-Ortiz (2000) Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.
  • N. Nielsen (1923) Traité Élémentaire des Nombres de Bernoulli. Gauthier-Villars, Paris.
  • N. E. Nörlund (1922) Mémoire sur les polynomes de Bernoulli. Acta Math. 43, pp. 121–196 (French).
  • M. Noumi and J. V. Stokman (2004) Askey-Wilson polynomials: an affine Hecke algebra approach. In Laredo Lectures on Orthogonal Polynomials and Special Functions, Adv. Theory Spec. Funct. Orthogonal Polynomials, pp. 111–144.