New fixed point results in controlled metric type spaces based on new contractive conditions

1.   Introduction

L. Carlitz (^[1]) introduced analogues of Bernoulli numbers for the rational function (finite) field $K = \mathbb F_r(T)$, which are called Bernoulli-Carlitz numbers now. Bernoulli-Carlitz numbers have been studied since then (e.g., see ^[2,3,4,5,6]). According to the notations by Goss ^[7], Bernoulli-Carlitz numbers $BC_n$ are defined by

Here, $e_C(x)$ is the Carlitz exponential defined by

where $D_i = [i][i-1]^r\cdots [1]^{r^{i-1}}$ ($i\ge 1$) with $D_0 = 1$, and $[i] = T^{r^i}-T$. The Carlitz factorial $\Pi(i)$ is defined by

for a non-negative integer $i$ with $r$-ary expansion:

As analogues of the classical Cauchy numbers $c_n$, Cauchy-Carlitz numbers $CC_n$ (^[8]) are introduced as

Here, $\log_C(x)$ is the Carlitz logarithm defined by

where $L_i = [i][i-1]\cdots [1]$ ($i\ge 1$) with $L_0 = 1$.

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.

On the other hand, for $N\ge 1$, hypergeometric Bernoulli numbers $B_{N, n}$ (^[9,10,11,12]) are defined by the generating function

where

is the confluent hypergeometric function with $(x)^{(n)} = x(x+1)\cdots(x+n-1)$ ($n\ge 1$) and $(x)^{(0)} = 1$. When $N = 1$, $B_n = B_{1, n}$ are classical Bernoulli numbers defined by

In addition, hypergeometric Cauchy numbers $c_{N, n}$ (see ^[13]) are defined by

where

is the Gauss hypergeometric function. When $N = 1$, $c_n = c_{1, n}$ are classical Cauchy numbers defined by

In ^[14], for $N\ge 0$, the truncated Bernoulli-Carlitz numbers $BC_{N, n}$ and the truncated Cauchy-Carlitz numbers $CC_{N, n}$ are defined by

and

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 $\alpha$ can be expressed uniquely as the simple continued fraction expansion:

where $a_0$ is an integer and $a_1, a_2, \dots$ 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 $a_0(x), a_1(x), \dots$ and $b_1(x), b_2(x), \dots$ are polynomials in $x$. In ^[16,17] several continued fraction expansions for non-exponential Bernoulli numbers are given. For example,

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 $B_{N, n}$ ($N\ge 1$, $n\ge 1$) 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].

In ^[21,22], the hypergeometric Cauchy numbers $c_{N, n}$ ($N\ge 1$, $n\ge 1$) can be expressed as

When $N = 1$, we have a determinant expression of Cauchy numbers ([20,p.50]).

Recently, in (^[23]) the truncated Euler-Carlitz numbers $EC_{N, n}$ ($N\ge 0$), 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 $N\ge 0$ and $n\ge 1$,

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.

2.   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:

with $P_{-1}(x) = 1$, $Q_{-1}(x) = 0$, $P_0(x) = a_0(x)$ and $Q_0(x) = 1$.

From the definition in (1.9), truncated Bernoulli-Carlitz numbers satisfy the relation

Thus,

yield that

Notice that the $n$-th convergent $p_n/q_n$ of the simple continued fraction (1.11) of a real number $\alpha$ yields the approximation property

Now,

and $P_n(x)$ and $Q_n(x)$ ($n\ge 2$) satisfy the recurrence relations

(They are proved by induction). Since by (2.2) for $n\ge 2$

we have the following continued fraction expansion.

Theorem 1.

Put $N = 0$ in Theorem 1 to illustrate a simpler case. Then, we have a continued fraction expansion concerning the original Bernoulli-Carlitz numbers.

Corollary 1.

From the definition in (1.10), truncated Cauchy-Carlitz numbers satisfy the relation

Thus,

yield that

Now,

and $P_n(x)$ and $Q_n(x)$ ($n\ge 2$) satisfy the recurrence relations

Since by (2.2) for $n\ge 2$

we have the following continued fraction expansion.

Theorem 2.

3.   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 $N\ge 0$ and $n\ge 1$,

[14,Theorem 2].

Now, we give a determinant expression of truncated Cauchy-Carlitz numbers.

Theorem 3. For integers $N\ge 0$ and $n\ge 1$,

where

with

We need the following Lemma in ^[27] in order to prove Theorem 3.

Lemma 1. Let $\{\alpha_n\}_{n\ge 0}$ be a sequence with $\alpha_0 = 1$, and $R(j)$ be a function independent of $n$. Then

if and only if

with $\alpha_0 = 1$.

Proof of Theorem 3. By the definition (1.10) with (1.6), we have

Thus, for $n\ge 1$, we get

By Lemma 1, we have

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 $r^{N+1}-r^N\nmid n$, since all the elements of one column (or row) become zero.

4.   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 $N\ge 0$ and $n\ge 1$,

[14,Theorem 1].

Now, we give a determinant expression of truncated Bernoulli-Carlitz numbers.

Theorem 4. For integers $N\ge 0$ and $n\ge 1$,

where

with $\delta_l^\ast$ 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 $r^{N+1}-r^N\nmid n$.

5.   Applications by the Trudi's formula

We shall use Trudi's formula to obtain different explicit expressions and inversion relations for the numbers $CC_{N, n}$ and $BC_{N, n}$.

Lemma 2. For a positive integer $n$, we have

where $\binom{t_1+\cdots + t_n}{t_1, \dots, t_n} = \frac{(t_1+\cdots + t_n)!}{t_1!\cdots t_n!}$ are the multinomial coefficients.

This relation is known as Trudi's formula [28,Vol.3,p.214], ^[29] and the case $a_0 = 1$ of this formula is known as Brioschi's formula ^[30], [28,Vol.3,pp.208–209].

In addition, there exists the following inversion formula (see, e.g. ^[27]), which is based upon the relation

Lemma 3. If $\{\alpha_n\}_{n\geq 0}$ is a sequence defined by $\alpha_0 = 1$ and

From Trudi's formula, it is possible to give the combinatorial expression

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.

Theorem 5. For integers $N\ge 0$ and $n\ge 1$, we have

where $a_n$ are given in Theorem 3.

Theorem 6. For integers $N\ge 0$ and $n\ge 1$, we have

where $d_n$ are given in Theorem 4.

By applying the inversion relation in Lemma 3 to Theorem 3 and Theorem 4, we have the following.

Theorem 7. For integers $N\ge 0$ and $n\ge 1$, we have

where $a_n$ is given in Theorem 3.

Theorem 8. For integers $N\ge 0$ and $n\ge 1$, we have

where $d_n$ is given in Theorem 4.

Acknowledgments

We would like to thank the referees for their valuable comments.

Conflict of interest

The authors declare no conflict of interest.