Continued fraction expansions of the generating functions of Bernoulli and related numbers
Section snippets
Continued fractions
Given an analytic function , several types of general continued fractions have been known and studied. J.H. Lambert expanded , and in continued fractions in 1768. Some special types are known as -fractions, -fractions, -fractions, -fractions and Hankel continued fractions (see, e.g., [7], [9], [16], [19]). Lando [14] discusses other similar continued fractions from the aspect of Combinatorics and generating functions. Loya [15, Chapter 8] exhibits intriguing ideas
Bernoulli numbers
Bernoulli numbers are defined by Many kinds of continued fraction expansions of the generating functions of Bernoulli numbers have been known and studied (see, e.g., [1, Appendix],[6]). However, those of generalized Bernoulli numbers seem to be few, though there exist several generalizations of the original Bernoulli numbers. In particular, direct generalizations from the continued fraction expansions seem to be hard.
Hypergeometric degenerate Bernoulli numbers are defined
Cauchy numbers
Cauchy numbers are defined by Note that these numbers are also called Gregory coefficients, reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind. In [4], new identities are found by using the continued fraction (see, e.g., [19, (90.1)]). In this section, we find a different type of continued fraction expansion related to Cauchy numbers.
Define the hypergeometric degenerate Cauchy numbers by the generating
Euler numbers
Hypergeometric Euler numbers [13] are defined by where is the hypergeometric function defined by When , then are the classical Euler numbers, defined by Since by (9) consider the convergents with Now,
Harmonic numbers
There are several generalizations of harmonic numbers , defined by In [20], the generalized harmonic numbers of order () are defined by where and are positive real numbers. When , are the -order harmonic numbers. When , are the original harmonic numbers.
Let () be the generalized rising factorial with . When ,
Functions associated with the Riemann zeta function
Let be Möbius function. From the property the Dirichlet series that generates the Möbius function is the multiplicative inverse of the Riemann zeta function: where is a complex number with real part larger than . Then we consider the convergents as Now, and and () satisfy the recurrence relations
Transforms of continued fractions
Raney [18] established a method to yield the simple continued fraction expansions of with from the simple continued fraction expansion where is an integer and are positive integers. In this section, we shall consider the linear fractional transformation for the continued fraction expansion in (1). Since the convergents () are given in (2) in our case, we have
Continued fractions of the ordinary generating functions
A continued fraction expansion of the ordinary generating function of Bernoulli numbers is given by ([1, A.5]). However, any beautiful continued fraction expansion for Cauchy numbers has not been known yet. Nevertheless, in order to satisfy the approximation property (3), we have the following expansion.
Theorem 8
Concluding remarks
In this paper, we deal with continued fraction in the aspects of convergents. Such techniques and ideas can be applied to more different types. For example, we can have a more complicated continued fraction expansion than that in Corollary 4.
Theorem 9 For ,
When , this is a direct generalization of the continued fraction expansion (see, e.g., [19, (90.1)]).
Similarly, the th convergent and the generating function of coincide up to the th term in their Taylor expansions. However, the structure
Acknowledgment
The author thanks the anonymous referee for his/her detailed comments and suggestions, which improved the quality of the paper.
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