Euler-Seidel matrices over Fp

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

A Euler--Seidel matrix is determined by an infinite sequence whose elements are given by recursion. The recurrence relations are investigated for numbers and polynomials such as hyperharmonics, Lucas numbers, and Euler and Genocchi polynomials. Linear recurring sequences in finite fields are employed, for instance, in coding theory and in several branches of electrical engineering. In this work, we define the period of a Euler--Seidel matrix over a field Fp with p elements, where p is a prime number. We give some results for the matrix whose initial sequence is \{sr(n)\}n=0\infty, where sr(n)=\sumk=0n {\binom{n}{k}}r, n \geq 0, and r is a fixed positive number. The numbers sr(n) play an important role in combinatorics and number theory. These numbers are known as Franel numbers for r=3.

Anahtar Kelimeler:

Euler--Seidel matrix

Euler-Seidel matrices over Fp

Keywords:

Euler--Seidel matrix,

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1] Calkin, N. Factors of sums of powers of binomial coefficients. Acta Arith. 86, 17–26 (1998).
[2] Dil, A., Mez˝o, I. Symmetric algorithm for hyperharmonic and Fibonacci Numbers. Appl. Math. Comput. 206, 942–951 (2008).
[3] Dumont, D. Matrices d’Euler-Seidel. Seminaire Lotharingien de Combinatorie. B05c 1981.
[4] Euler, L. De transformatione serierum. Opera Omnia 1: Teubner 1913. [5] Franel, J. On a question of Laisant. L’interm´ediaire des Math´ematiciens 1, 45–47 (1894).
[6] Franel, J. On a question of J. Franel. L’interm´ediaire des Math´ematiciens 2, 33–35 (1895).
[7] Seidel, P. Uber eine einfache Entstehungsweise der Bernoullishen Zahlen und einiger verwandten. Sitzungsberichte ¨ der M¨unh. Akad. Math. Phys. 7, 157–187 (1877).
[8] Yuan, J., Lu, Z., Schmidt, A. On recurrences for sums of powers of binomial coefficients. J. Number Theory 128, 2784–2794 (2008).