Gregory coefficients

Last updated

Gregory coefficientsGn, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind, [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] are the rational numbers that occur in the Maclaurin series expansion of the reciprocal logarithm

Contents

Gregory coefficients are alternating Gn = (−1)n−1|Gn| for n > 0 and decreasing in absolute value. These numbers are named after James Gregory who introduced them in 1670 in the numerical integration context. They were subsequently rediscovered by many mathematicians and often appear in works of modern authors, who do not always recognize them. [1] [5] [14] [15] [16] [17]

Numerical values

n1234567891011... OEIS sequences
Gn+1/21/12+1/2419/720+3/160863/60480+275/2419233953/3628800+8183/10368003250433/479001600+4671/788480... OEIS:  A002206 (numerators),

OEIS:  A002207 (denominators)

Computation and representations

The simplest way to compute Gregory coefficients is to use the recurrence formula

with G1 = 1/2. [14] [18] Gregory coefficients may be also computed explicitly via the following differential

or the integral

which can be proved by integrating between 0 and 1 with respect to , once directly and the second time using the binomial series expansion first.

It implies the finite summation formula

where s(n,) are the signed Stirling numbers of the first kind.

and Schröder's integral formula [19] [20]

Bounds and asymptotic behavior

The Gregory coefficients satisfy the bounds

given by Johan Steffensen. [15] These bounds were later improved by various authors. The best known bounds for them were given by Blagouchine. [17] In particular,

Asymptotically, at large index n, these numbers behave as [2] [17] [19]

More accurate description of Gn at large n may be found in works of Van Veen, [18] Davis, [3] Coffey, [21] Nemes [6] and Blagouchine. [17]

Series with Gregory coefficients

Series involving Gregory coefficients may be often calculated in a closed-form. Basic series with these numbers include

where γ = 0.5772156649... is Euler's constant. These results are very old, and their history may be traced back to the works of Gregorio Fontana and Lorenzo Mascheroni. [17] [22] More complicated series with the Gregory coefficients were calculated by various authors. Kowalenko, [8] Alabdulmohsin [10] [11] and some other authors calculated

Alabdulmohsin [10] [11] also gives these identities with

Candelperger, Coppo [23] [24] and Young [7] showed that

where Hn are the harmonic numbers. Blagouchine [17] [25] [26] [27] provides the following identities

where li(z) is the integral logarithm and is the binomial coefficient. It is also known that the zeta function, the gamma function, the polygamma functions, the Stieltjes constants and many other special functions and constants may be expressed in terms of infinite series containing these numbers. [1] [17] [18] [28] [29]

Generalizations

Various generalizations are possible for the Gregory coefficients. Many of them may be obtained by modifying the parent generating equation. For example, Van Veen [18] consider

and hence

Equivalent generalizations were later proposed by Kowalenko [9] and Rubinstein. [30] In a similar manner, Gregory coefficients are related to the generalized Bernoulli numbers

see, [18] [28] so that

Jordan [1] [16] [31] defines polynomials ψn(s) such that

and call them Bernoulli polynomials of the second kind . From the above, it is clear that Gn = ψn(0). Carlitz [16] generalized Jordan's polynomials ψn(s) by introducing polynomials β

and therefore

Blagouchine [17] [32] introduced numbers Gn(k) such that

obtained their generating function and studied their asymptotics at large n. Clearly, Gn = Gn(1). These numbers are strictly alternating Gn(k) = (-1)n-1|Gn(k)| and involved in various expansions for the zeta-functions, Euler's constant and polygamma functions. A different generalization of the same kind was also proposed by Komatsu [31]

so that Gn = cn(1)/n! Numbers cn(k) are called by the author poly-Cauchy numbers. [31] Coffey [21] defines polynomials

and therefore |Gn| = Pn+1(1).

See also

Related Research Articles

<span class="mw-page-title-main">Binomial coefficient</span> Number of subsets of a given size

In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers nk ≥ 0 and is written It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n; this coefficient can be computed by the multiplicative formula

In mathematics, the Bernoulli numbersBn are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function has a root at , then , taking the limit value at , is an entire function. On the other hand, the natural logarithm, the reciprocal function, and the square root are all not entire functions, nor can they be continued analytically to an entire function.

<span class="mw-page-title-main">Gamma function</span> Extension of the factorial function

In mathematics, the gamma function is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n,

<span class="mw-page-title-main">Riemann zeta function</span> Analytic function in mathematics

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined as for , and its analytic continuation elsewhere.

<span class="mw-page-title-main">Euler's constant</span> Constant value used in mathematics

Euler's constant is a mathematical constant, usually denoted by the lowercase Greek letter gamma, defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log:

<span class="mw-page-title-main">Harmonic number</span> Sum of the first n whole number reciprocals; 1/1 + 1/2 + 1/3 + ... + 1/n

In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:

<span class="mw-page-title-main">Polygamma function</span> Meromorphic function

In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers defined as the (m + 1)th derivative of the logarithm of the gamma function:

<span class="mw-page-title-main">Digamma function</span> Mathematical function

In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:

<span class="mw-page-title-main">Polylogarithm</span> Special mathematical function

In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Lis(z) of order s and argument z. Only for special values of s does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polylogarithm function appears as the closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein integral. In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams.

The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight. Euler generalised the problem considerably, and his ideas were taken up more than a century later by Bernhard Riemann in his seminal 1859 paper "On the Number of Primes Less Than a Given Magnitude", in which he defined his zeta function and proved its basic properties. The problem is named after Basel, hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the problem.

<span class="mw-page-title-main">Stieltjes constants</span> Constants in the zeta functions Laurent series expansion

In mathematics, the Stieltjes constants are the numbers that occur in the Laurent series expansion of the Riemann zeta function:

In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles.

In mathematics, the Riemann zeta function is a function in complex analysis, which is also important in number theory. It is often denoted and is named after the mathematician Bernhard Riemann. When the argument is a real number greater than one, the zeta function satisfies the equation It can therefore provide the sum of various convergent infinite series, such as Explicit or numerically efficient formulae exist for at integer arguments, all of which have real values, including this example. This article lists these formulae, together with tables of values. It also includes derivatives and some series composed of the zeta function at integer arguments.

<span class="mw-page-title-main">Carl Johan Malmsten</span> Swedish mathematician and politician (1814–1886)

Carl Johan Malmsten was a Swedish mathematician and politician. He is notable for early research into the theory of functions of a complex variable, for the evaluation of several important logarithmic integrals and series, for his studies in the theory of Zeta-function related series and integrals, as well as for helping Mittag-Leffler start the journal Acta Mathematica. Malmsten became Docent in 1840, and then, Professor of mathematics at the Uppsala University in 1842. He was elected a member of the Royal Swedish Academy of Sciences in 1844. He was also a minister without portfolio in 1859–1866 and Governor of Skaraborg County in 1866–1879.

In mathematics, the secondary measure associated with a measure of positive density ρ when there is one, is a measure of positive density μ, turning the secondary polynomials associated with the orthogonal polynomials for ρ into an orthogonal system.

In mathematics, the Mittag-Leffler polynomials are the polynomials gn(x) or Mn(x) studied by Mittag-Leffler (1891).

The Bernoulli polynomials of the second kindψn(x), also known as the Fontana–Bessel polynomials, are the polynomials defined by the following generating function:

References

  1. 1 2 3 4 Ch. Jordan. The Calculus of Finite Differences Chelsea Publishing Company, USA, 1947.
  2. 1 2 L. Comtet. Advanced combinatorics (2nd Edn.) D. Reidel Publishing Company, Boston, USA, 1974.
  3. 1 2 H.T. Davis. The approximation of logarithmic numbers. Amer. Math. Monthly, vol. 64, no. 8, pp. 11–18, 1957.
  4. P. C. Stamper. Table of Gregory coefficients. Math. Comp. vol. 20, p. 465, 1966.
  5. 1 2 D. Merlini, R. Sprugnoli, M. C. Verri. The Cauchy numbers. Discrete Math., vol. 306, pp. 1906–1920, 2006.
  6. 1 2 G. Nemes. An asymptotic expansion for the Bernoulli numbers of the second kind. J. Integer Seq, vol. 14, 11.4.8, 2011
  7. 1 2 P.T. Young. A 2-adic formula for Bernoulli numbers of the second kind and for the Nörlund numbers. J. Number Theory, vol. 128, pp. 2951–2962, 2008.
  8. 1 2 V. Kowalenko. Properties and Applications of the Reciprocal Logarithm Numbers. Acta Appl. Math., vol. 109, pp. 413–437, 2010.
  9. 1 2 V. Kowalenko. Generalizing the reciprocal logarithm numbers by adapting the partition method for a power series expansion. Acta Appl. Math., vol. 106, pp. 369–420, 2009.
  10. 1 2 3 Alabdulmohsin, Ibrahim M. (2012). "Summability Calculus". arXiv: 1209.5739 [math.CA].
  11. 1 2 3 Alabdulmohsin, Ibrahim M. (2018). Summability Calculus. doi:10.1007/978-3-319-74648-7. ISBN   978-3-319-74647-0.
  12. F. Qi and X.-J. Zhang An integral representation, some inequalities, and complete monotonicity of Bernoulli numbers of the second kind. Bull. Korean Math. Soc., vol. 52, no. 3, pp. 987–98, 2015.
  13. Weisstein, Eric W. "Logarithmic Number." From MathWorld—A Wolfram Web Resource.
  14. 1 2 J. C. Kluyver. Euler's constant and natural numbers. Proc. K. Ned. Akad. Wet., vol. 27(1-2), 1924.
  15. 1 2 J.F. Steffensen. Interpolation (2nd Edn.). Chelsea Publishing Company, New York, USA, 1950.
  16. 1 2 3 L. Carlitz. A note on Bernoulli and Euler polynomials of the second kind. Scripta Math., vol. 25, pp. 323–330,1961.
  17. 1 2 3 4 5 6 7 8 Ia.V. Blagouchine. Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to π1. J.Math. Anal. Appl., 2015.
  18. 1 2 3 4 5 S.C. Van Veen. Asymptotic expansion of the generalized Bernoulli numbersBn(n  1)for large values of n (n integer). Indag. Math. (Proc.), vol. 13, pp. 335–341, 1951.
  19. 1 2 I. V. Blagouchine, A Note on Some Recent Results for the Bernoulli Numbers of the Second Kind, Journal of Integer Sequences, Vol. 20, No. 3 (2017), Article 17.3.8 arXiv:1612.03292
  20. Ernst Schröder, Zeitschrift fur Mathematik und Physik, vol. 25, pp. 106–117 (1880)
  21. 1 2 M.W. Coffey. Series representations for the Stieltjes constants. Rocky Mountain J. Math., vol. 44, pp. 443–477, 2014.
  22. Blagouchine, Iaroslav V. (2015). "A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations". Journal of Number Theory. 148: 537–592. arXiv: 1401.3724 . doi:10.1016/j.jnt.2014.08.009.
  23. Candelpergher, Bernard; Coppo, Marc-Antoine (2012). "A new class of identities involving Cauchy numbers, harmonic numbers and zeta values". The Ramanujan Journal. 27 (3): 305–328. doi:10.1007/s11139-011-9361-7.
  24. B. Candelpergher and M.-A. Coppo. A new class of identities involving Cauchy numbers, harmonic numbers and zeta values. Ramanujan J., vol. 27, pp. 305–328, 2012
  25. OEIS:  A269330
  26. OEIS:  A270857
  27. OEIS:  A270859
  28. 1 2 N. Nörlund. Vorlesungen über Differenzenrechnung. Springer, Berlin, 1924.
  29. Ia.V. Blagouchine. Expansions of generalized Euler's constants into the series of polynomials in π2 and into the formal enveloping series with rational coefficients only J. Number Theory, vol. 158, pp. 365–396, 2016.
  30. Rubinstein, Michael O. (2012). "Identities for the Riemann zeta function". The Ramanujan Journal. 27: 29–42. doi:10.1007/s11139-010-9276-8.
  31. 1 2 3 "Takao Komatsu. On poly-Cauchy numbers and polynomials, 2012" (PDF). Archived from the original (PDF) on 2016-03-16. Retrieved 2016-05-20.
  32. Ia.V. Blagouchine. Three Notes on Ser's and Hasse's Representations for the Zeta-functions Integers (Electronic Journal of Combinatorial Number Theory), vol. 18A, Article #A3, pp. 1–45, 2018. arXiv:1606.02044