Hagen–Rothe convolution identities through Lagrange interpolations

New proofs of Hagen–Rothe identities concerning binomial convolutions are presented through Lagrange interpolations. Keywords: Chu–Vandermonde formula; Hagen–Rothe identities; ﬁnite diﬀerence; Lagrange interpolation. 2020 Mathematics Subject Classiﬁcation: 05A10, 11B65.


Introduction and Outline
The Chu-Vandermonde convolution formula is one of the most popular binomial identities: n k=0 x k y n − k = x + y n .
The first two were discovered by Hagen and Rothe (cf. Comtet [10, §3.1] and Graham et al. [14, §5.4]), while the third identity is due to Jensen [15]. Their limiting cases result in the well-known Abel identities (cf. [1,8]). When λ = 0, all of them become the Chu-Vandermonde formula. Analogously for λ = 1, it is routine to check that the corresponding formulae are equivalent to the Chu-Vandermonde formula. These convolution identities have wide applications to enumerative combinatorics and number theory. The reader can refer to Strehl [20] for a historical note. The typical proofs for these convolution identities are highlighted as follows: • Generating function method: Gould [12,13] (see Chu [3] also).
• Gould-Hsu inverse series relations: Chu and Hsu [2,9]. • Riordan arrays (equivalent to the Lagrange expansion formula): Sprugnoli [19]. * E-mail address: chu.wenchang@unisalento.it The objective of this article is to present proofs of Identities (1), (2), and (3), by means of Lagrange interpolations and finite difference method. Let ∆ be the difference operator with unit increment For a natural number n, the differences of order n is given by which is expressed by the following Newton-Gregory formula For an indeterminate x and an integer n, recall that the Pochhammer symbol (x) n is defined by When p m (x) is a polynomial of degree m ≤ n with the leading coefficient c m , the following properties (cf. [4][5][6]8]) are quite useful: where χ stands for the logical function with χ(true) = 1 and χ(false) = 0. The former equality is well-known. The latter can be justified easily as follows. First when p m (x) ≡ 1, it is trivial to check it by the induction principle. Observing that In addition, we fix ∆ n 0 f (x) = ∆ n f (x) x=0 for the differences starting at x = 0.

Proofs of three formulae
Now, we are in a position to present detailed proofs for the three identities announced in the introduction. To be precise, we assume that λ is a variable subject to λ = 1 throughout this section, even though three identities (1), (2) and (3) are also valid in this case as declared in the introduction. (1) Denote by P(x) the binomial sum in (1), which is obviously a polynomial of degree n. Its value at x = m − y can be reformulated as

Proof of Identity
By making use of the two equalities we have the following alternative expression Then P(m − y) vanishes for 0 ≤ m < n, because it results in the nth difference of the polynomial (xλ − y) m (1 + m − x − y + xλ) n−m−1 of degree n − 1. When m = n, we can evaluate Therefore, the first convolution identity (1) follows from the Lagrange polynomial of P(x) for the interpolation points {x m := m − y} n m=0 , which contains the only surviving term with m = n: (2) Let Q(x) be the binomial sum in (2), which is again a polynomial of degree n. Its value at x = m − y reads as

Proof of Identity
For 1 ≤ m < n, it is clear that Q(m−y) vanishes because it results in the nth difference of the polynomial (1−y +xλ) m−1 (1+ m − x − y + xλ) n−m−1 of degree n − 2. In addition, we can further evaluate Consequently, the second convolution identity (2) follows from the Lagrange polynomial of Q(x) for the interpolation points {x m := m − y} n m=0 , which consists of the only two terms corresponding to m = 0 and m = n: (3) Assume that R(x) stands for the left sum of (3). Its value at x = m − y can be expressed as

Proof of Identity
When 0 ≤ m < n, we have that R(m − y) results in the nth difference of the polynomial (xλ − y) which coincides with the right sum of (3) at x = m − y: Furthermore, we can evaluate This coincides again with the right sum of (3) at x = n − y: This not only confirms Jensen's convolution identity (3), but also gives, as a bonus, the third expression