On hypergeometric Cauchy numbers of higher grade

Abstract: In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy, and Euler numbers. Cauchy numbers can be generalized to the hypergeometric Cauchy numbers. Recently, Barman et al. study more general numbers in terms of determinants, which involve Bernoulli, Euler and Lehmer’s generalized Euler numbers. However, Cauchy numbers and their generalizations are not involved in these generalized numbers. In this paper, we study more general numbers in terms of determinants, which involve Cauchy numbers. The motivations and backgrounds of the definition are in an operator related to graph theory. We also give several expressions and identities by Trudi’s and inversion formulae.

A generalized version for Bernoulli and Euler numbers has been established in [17], where the elements contain factorials, as seen in (1.8), (1.9), (1.10) and (1.7). However, expressions for Cauchy and their generalized numbers cannot be included because they do not contain the factorial elements, as seen in (1.13). Universal Bernoulli numbers were studied in [1] and [8], and particularly, some universal Kummer congruences were established in [1] and [8].
In this paper, we introduce the hypergeometric Cauchy numbers of higher grade that are introduced as generalizations of both hypergeometric Cauchy numbers and the classical Cauchy numbers. We give several expressions and identities.

Hypergeometric Cauchy numbers of higher grade
For N ≥ 1 and n ≥ 0, define hypergeometric Cauchy numbers V ( j) N,n,r ( j = 0, 1) of grade r by where 2 F 1 (a, b; c; z) is the Gauss hypergeometric function, defined by From the definition, V ( j) N,n,r ≡ 0 (mod r) unless n ≡ 0 (mod r). When r = 1 and j = 0 in (2.1), c N,n = V (0) N,n,1 are the hypergeometric Cauchy numbers in (1.11). When N = 1, r = 1 and j = 0 in (2.1), c n = V (0) 1,n,1 are the classical Cauchy numbers in (1.12). We can write (2.1) as The definition (2.1) with (2.2) may be obvious or artificial for the readers with different backgrounds. However, our initial motivations were from Combinatorics, in particular, graph theory. In 1989, Cameron [5] considered the operator A defined on the set of sequences of non-negative integers as follows: for Cameron's operators deal with only nonnegative integers, but the operators can be used extensively for rational numbers. In the sense of Cameron's operator A, we have the following relation.
This relation is interchangeable in the sense of determinants too. See Section 5 about Trudi's formula.
We have the following recurrence relation.
Comparing the coefficient on both sides, we obtain We have an explicit expression of V ( j) N,n,r . Theorem 1. Let j = 0, 1. For n ≥ 1, Proof. The proof is done by induction on n. From Proposition 1 with n = 1, This matches the result when n = 1. Assume that the result is valid up to n − 1. Then by Proposition 1 There is an alternative form of V ( j) N,n,r by using binomial coefficients. The proof is similar to that of Theorem 1 and is omitted.

Determinantal expressions
In this section, we shall show an expression of hypergeometric Cauchy numbers of higher grade in terms of determinants. This result is a generalization of those of the hypergeometric and the classical Cauchy numbers. For simplification of determinant expressions, we use the Jordan matrix J 0 is the identity matrix and J T is the transpose matrix of J.
Theorem 3. For n ≥ 1, Proof. For simplicity, put V N,n = V ( j) N,n,r /n!. Then, we shall prove that for any n ≥ 1 When n = 1, (3.1) is valid because by Theorem 1 we get Assume that (3.1) is valid up to n − 1. Notice that by Proposition 1, we have Thus, by expanding the right-hand side of (3.1) along the first row, it is equal to Remark. When r = 1 and j = 0, the determinant expression in Theorem 3 is reduced to that in (1.13) for the hypergeometric Cauchy numbers c N,n = V (0) N,n,1 . When N = 1, r = 1 and j = 0, we have a determinant expression of the Cauchy numbers c n = V (0) 1,n,1 ([6, p.50]).

Incomplete hypergeometric Cauchy numbers of higher grade
As applications or variations to generalize the hypergeometric numbers V ( j) N,n,r of higher grade, we shall introduce two kinds of incomplete hypergeometric Cauchy numbers of higher grade. Similar but slightly different kinds of incomplete numbers are considered in [10,12,14,17]. In addition, similar techniques can be found in [24] and later cited in [7]. For j = 0, 1 and n ≥ m ≥ 1, define the restricted hypergeometric Cauchy numbers V ( j) N,n,r,≤m of grade r by When m → ∞ in (4.1) and m = 1 in (4.2), V ( j) N,n,r = V ( j) N,n,r,≤∞ = V ( j) N,n,r,≥1 are the original hypergeometric Cauchy numbers of grade r, defined in (2.1) with (2.2). Hence, both incomplete numbers are reduced to the hypergeometric Cauchy numbers too.
Notice that V ( j) N,n,r,≤m = V ( j) N,n,r,≥m = 0 unless n ≡ 0 (mod r). The restricted and associated hypergeometric Cauchy numbers satisfy the following recurrence relations. Proof. First, we shall prove the relation for the restricted hypergeometric Cauchy numbers. By the definition (4.1), we get Comparing the coefficient on both sides, we obtain the first identity. Next, we prove the relation for the associated hypergeometric Cauchy numbers. By the definition (4.2), we get Comparing the coefficient on both sides, we obtain the desired result.
The restricted and associated hypergeometric Cauchy numbers have the following expressions in terms of determinants. From the expression of Theorem 3, all the elements change to 0 in more diagonal directed bands. Proof. First, we shall prove the first expression for the restricted hypergeometric Cauchy numbers. For simplicity, put V N,rn,≤m = V ( j) N,rn,r,≤m /(rn)! and prove that for n ≥ m ≥ 1 When n = m, we have V N,rm,≤m = V N,rm , and the result reduces to Theorem 3. Assume that (4.3) is valid up to n − 1. If n ≥ 2m, then the determinant on the right-hand side of (4.3) is equal to If m < n ≤ 2m, then the determinant on the right-hand side of (4.3) is equal to · · · rN+ j rN+r+ j = V N,rn−r,≤m (rN + j) rN + r + j − V N,rn−2r,≤m (rN + j) rN + 2r + j + · · · + (−1) n−m−1 V N,rm,≤m (rN + j) rN + rn − rm + j = V N,rn−r,≤m (rN + j) rN + r + j − V N,rn−2r,≤m (rN + j) rN + 2r + j + · · · + (−1) m−1 V N,rn−rm,≤m (rN + j) rN + rm + j = V N,rn,≤m .
Next, we prove the second expression for the associated hypergeometric Cauchy numbers. For simplicity, put V N,rn,≥m = V N,rn,r,≥m /(rn)! and we prove that If m ≤ n ≤ 2m, the determinant on the right-hand side of (4.4) is equal to  Since only the term for k = 0 does not vanish in the second relation of Proposition 2, we have V N,rn,≥m = (−1) n+1 rN + j rN + rn + j .
If n ≥ 2m, the determinant on the right-hand side of (4.4) is equal to Here, we used the second relation of Proposition 2 again.
For n, m ≥ 1, Proof. First, we shall prove the first expression for the restricted hypergeometric Cauchy numbers. When n ≤ m, the proof is similar to that of Proposition 1. Note that in the proof of Proposition 1, Let n ≥ m + 1. By the first relation of Proposition 2 (rN + j) k−1 (rN + ri 1 + j) · · · (rN + ri k−1 + j) .
By putting i k = n−l, in the first term by n−1 ≥ l ≥ k−1 ≥ n−m, in the second term by n−1 ≥ l ≥ n−m, Therefore, .
By applying the inversion relation in Lemma 2 to Theorem 3, we have the following. In this sense, we have the inversion relation of Corollary 1 too.

Conclusions
In this paper, we proposed one type of generalizations of the classical Cauchy numbers and hypergeometric Cauchy numbers. Many other generalizations are known, but the focus of this paper is on the determinant, which originated in Glaisher and others. Similar determinants have been dealt with by Brioshi, Trudi and others, but have long been forgotten. A similar generalization attempt, made by the first author of this paper with Barman in 2019, has proposed generalized numbers including the classical Bernoulli numbers, hypergeometric Bernoulli numbers, Euler numbers, hypergeometric Euler numbers, and so on. However, classical Cauchy numbers and hypergeometric Cauchy numbers cannot be included in the generalization by Barman et al., and this is achieved in this paper. The background and motivation for generalization is Cameron's operator, which is related to graph theory. There, only integers were targeted, but in this paper, we extended this to rational numbers and applied it.