Several expressions of truncated Bernoulli-Carlitz and truncated Cauchy-Carlitz numbers

Abstract: The truncated Bernoulli-Carlitz numbers and the truncated Cauchy-Carlitz numbers are defined as analogues of hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers, and as extensions of Bernoulli-Carlitz numbers and the Cauchy-Carlitz numbers. These numbers can be expressed explicitly in terms of incomplete Stirling-Carlitz numbers. In this paper, we give several expressions of truncated Bernoulli-Carlitz numbers and truncated Cauchy-Carlitz numbers as natural extensions. One kind of expressions is in continued fractions. Another is in determinants originated in Glaisher, giving several interesting determinant expressions of numbers, including Bernoulli and Cauchy numbers.

In [8], Bernoulli-Carlitz numbers and Cauchy-Carlitz numbers are expressed explicitly by using the Stirling-Carlitz numbers of the second kind and of the first kind, respectively. These properties are the extensions that Bernoulli numbers and Cauchy numbers are expressed explicitly by using the Stirling numbers of the second kind and of the first kind, respectively.
In addition, hypergeometric Cauchy numbers c N,n (see [13]) are defined by is the Gauss hypergeometric function. When N = 1, c n = c 1,n are classical Cauchy numbers defined by c n x n n! .
In [14], for N ≥ 0, the truncated Bernoulli-Carlitz numbers BC N,n and the truncated Cauchy-Carlitz numbers CC N,n are defined by respectively. When N = 0, BC n = BC 0,n and CC n = CC 0,n are the original Bernoulli-Carlitz numbers and Cauchy-Carlitz numbers, respectively. These numbers BC N,n and CC N,n in (1.9) and (1.10) in function fields are analogues of hypergeometric Bernoulli numbers in (1.7) and hypergeometric Cauchy numbers in (1.8) in complex numbers, respectively. In [15], the truncated Euler polynomials are introduced and studied in complex numbers. It is known that any real number α can be expressed uniquely as the simple continued fraction expansion: where a 0 is an integer and a 1 , a 2 , . . . are positive integers. Though the expression is not unique, there exist general continued fraction expansions for real or complex numbers, and in general, analytic functions f (x): where a 0 (x), a 1 (x), . . . and b 1 (x), b 2 (x), . . . are polynomials in x. In [16,17] several continued fraction expansions for non-exponential Bernoulli numbers are given. For example, (1.13) More general continued fractions expansions for analytic functions are recorded, for example, in [18].
In this paper, we shall give expressions for truncated Bernoulli-Carlitz numbers and truncated Cauchy-Carlitz numbers.
In [19], the hypergeometric Bernoulli numbers B N,n (N ≥ 1, n ≥ 1) can be expressed as When N = 1, we have a determinant expression of Bernoulli numbers ( [20, p.53]). In addition, relations between B N,n and B N−1,n are shown in [19].
Recently, in ( [23]) the truncated Euler-Carlitz numbers EC N,n (N ≥ 0), introduced as are shown to have some determinant expressions. When N = 0, EC n = EC 0,n are the Euler-Carlitz numbers, denoted by where is the Carlitz hyperbolic cosine. This reminds us that the hypergeometric Euler numbers E N,n ( [24]), defined by have a determinant expression [25,Theorem 2.3] for N ≥ 0 and n ≥ 1, When N = 0, we have a determinant expression of Euler numbers (cf. [20, p.52]). More general cases are studied in [26].
In this paper, we also give similar determinant expressions of truncated Bernoulli-Carlitz numbers and truncated Cauchy-Carlitz numbers as natural extensions of those of hypergeometric numbers.

Continued fraction expansions of truncated Bernoulli-Carlitz and Cauchy-Carlitz numbers
Let the n-th convergent of the continued fraction expansion of (1.12) be .
There exist the fundamental recurrence formulas: From the definition in (1.9), truncated Bernoulli-Carlitz numbers satisfy the relation Thus, Notice that the n-th convergent p n /q n of the simple continued fraction (1.11) of a real number α yields the approximation property and P n (x) and Q n (x) (n ≥ 2) satisfy the recurrence relations (They are proved by induction). Since by (2.2) for n ≥ 2 we have the following continued fraction expansion.
Put N = 0 in Theorem 1 to illustrate a simpler case. Then, we have a continued fraction expansion concerning the original Bernoulli-Carlitz numbers. .

From the definition in (1.10), truncated Cauchy-Carlitz numbers satisfy the relation
Thus, and P n (x) and Q n (x) (n ≥ 2) satisfy the recurrence relations Since by (2.2) for n ≥ 2 a n (x) = L N+n L N+n−1 − x r N+n +r N+n−1 and b n (x) = L N+n−1 L N+n−2 x r N+n −r N+n−1 , we have the following continued fraction expansion.

A determinant expression of truncated Cauchy-Carlitz numbers
In [14], some expressions of truncated Cauchy-Carlitz numbers have been shown. One of them is for integers N ≥ 0 and n ≥ 1,  −a 1 1 (−1) n a n · · · · · · a 2 −a 1 , where We need the following Lemma in [27] in order to prove Theorem 3.
Lemma 1. Let {α n } n≥0 be a sequence with α 0 = 1, and R( j) be a function independent of n. Then .
Examples. When n = r N+1 − r N , Let n = r N+2 − r N . For simplicity, put Then by expanding at the first column, we have The second term is equal to The first term is Therefore, From this procedure, it is also clear that CC N,n = 0 if r N+1 − r N n, since all the elements of one column (or row) become zero.

A determinant expression of truncated Bernoulli-Carlitz numbers
In [14], some expressions of truncated Bernoulli-Carlitz numbers have been shown. One of them is for integers N ≥ 0 and n ≥ 1, with δ * l as in (3.2). Proof. The proof is similar to that of Theorem 3, using (1.9) and (1.2).
Example. Let n = 2(r N+1 − r N ). For convenience, put Then, we have
It is also clear that BC N,n = 0 if r N+1 − r N n.
By applying these lemmata to Theorem 3 and Theorem 4, we obtain an explicit expression for the truncated Cauchy-Carlitz numbers and the truncated Bernoulli-Carlitz numbers.