Three Determinant Evaluations
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Three Determinant Evaluations Wenchang Chu1,2 · Emrah Kılıç3 Received: 20 May 2020 / Revised: 4 August 2020 / Accepted: 17 August 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract By means of matrix decompositions, three determinants with their entries being binomial sums are evaluated in closed forms. Ten remarkable examples are illustrated as propositions, which present determinant identities about binomial coefficients and quotients of rising factorials, as well as orthogonal polynomials named after Hermite, Laguerre and Legendre. Keywords Determinant · Decomposition · Factorial · Binomial coefficient Mathematics Subject Classification 15A15 · 15A23 · 05A10
1 Introduction There exist numerous interesting determinant evaluations, whose matrix entries are well-known combinatorial number sequences (cf. [1–5]), for example, binomial coefficients, rising and falling factorials, as well as their reciprocals. They are proved mainly by means of generating functions and combinatorial enumerations (cf. [6,7]), finite difference method (cf. [8,9]) and matrix decompositions (cf. [10–13]). The reader may refer to [14] for a comprehensive coverage.
Communicated by Rosihan M. Ali.
B
Emrah Kılıç [email protected] Wenchang Chu [email protected]
1
School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou, Henan, People’s Republic of China
2
Present Address: Department of Mathematics and Physics, University of Salento, (P. O. Box 193), 73100 Lecce, Italy
3
Mathematics Department, TOBB University of Economics and Technology, Sögütözü 06560, Ankara, Turkey
123
W. Chu, E. Kılıç
Recently, Ekhad and Zeilberger [15] considered the following matrix K n = K n (i, j) 0≤i, j≤n :
K n (i, j) =
j k=0
(−1)k
i n−i 2 k j −k
and determined explicitly its spectrum. This motivates us further to examine three extended matrices with their entries being binomial sums and containing many free parameters. In the next section, we shall evaluate the first determinant (see Theorem 1) by the coefficient extraction method. Then, the rest of the paper will be devoted to the other two matrices. By expressing them as products of a matrix of Vandermonde type and an upper triangular one, we shall explicitly evaluate their determinants (see Theorems 2 and 8) as products including factors of binomial sums with the σ -sequence. By specifying concretely the σ -sequence, we derive, from these two determinant evaluations, ten remarkable determinant identities (highlighted as propositions), involving binomial coefficients and quotients of rising factorials, as well as orthogonal polynomials named after Hermite, Laguerre and Legendre.
2 The First Determinant As a warm up, we start to evaluate an easy determinant by means of the method of extracting coefficients. Denote by [T k ]φ(T ) the coefficient of T k in the formal power series φ(T ). For the matrix A defined by A = ai, j 0≤i, j≤n :
ai, j =
j x +i y−i , (−1)k k j −k k=0
we can express it as de
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