Gregory coefficients, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind, are the rational numbers that occur in the Maclaurin series expansion of the reciprocal logarithm Gregory coefficients are alternating 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.
The Gregory coefficients satisfy the bounds given by Johan Steffensen. These bounds were later improved by various authors. The best known bounds for them were given by Blagouchine. In particular, Asymptotically, at large index, these numbers behave as More accurate description of at large may be found in works of Van Veen, Davis, Coffey, Nemes and Blagouchine.
Series with Gregory coefficients
Series involving Gregory coefficients may be often calculated in a closed-form. Basic series with these numbers include where 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. More complicated series with the Gregory coefficients were calculated by various authors. Kowalenko, Alabdulmohsin and some other authors calculated Alabdulmohsin also gives these identities Candelperger, Coppo and Young showed that where are the harmonic numbers. Blagouchine provides the following identities where 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.
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 consider and hence Equivalent generalizations were later proposed by Kowalenko and Rubinstein. In a similar manner, Gregory coefficients are related to the generalized Bernoulli numbers see, so that Jordan defines polynomials such that and call them Bernoulli polynomials of the second kind. From the above, it is clear that. Carlitz generalized Jordan's polynomials by introducing polynomials and therefore Blagouchine introduced numbers such that obtained their generating function and studied their asymptotics at large. Clearly,. These numbers are strictly alternating 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 so that Numbers are called by the author poly-Cauchy numbers. Coffey defines polynomials and therefore.
