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Representation numbers of five sextenary quadratic forms

Ernest X. W. Xia, Olivia X. M. Yao, A. F. Y. Zhao (2015)

Colloquium Mathematicae

For nonnegative integers a, b, c and positive integer n, let N(a,b,c;n) denote the number of representations of n by the form ∑ i = 1 a ( x ² i + x i y i + y ² i ) + 2 ∑ j = 1 b ( u ² j + u j v j + v ² j ) + 4 ∑ k = 1 c ( r ² k + r k s k + s ² k ) . Explicit formulas for N(a,b,c;n) for some small values were determined by Alaca, Alaca and Williams, by Chan and Cooper, by Köklüce, and by Lomadze. We establish formulas for N(2,1,0;n), N(2,0,1;n), N(1,2,0;n), N(1,0,2;n) and N(1,1,1;n) by employing the (p, k)-parametrization of three 2-dimensional theta functions due to Alaca, Alaca and Williams.

Résidus de puissances et formes quadratiques

Dominique Bernardi (1980)

Annales de l'institut Fourier

Pour tout entier q et certains entiers n , les nombres premiers p - congrus à 1 modulo n - tels que q soit le résidu d’une puissance n -ième modulo p sont caractérisés par le fait que certains systèmes de φ ( n ) / 2 formes quadratiques à coefficients entiers en φ ( n ) variables représentent le φ ( n ) / 2 -uplet ( p , 0 , 0 ... , 0 ) . La démonstration de ce résultat est accompagnée d’une méthode explicite de construction de ces systèmes.

Root location for the characteristic polynomial of a Fibonacci type sequence

Zhibin Du, Carlos Martins da Fonseca (2023)

Czechoslovak Mathematical Journal

We analyse the roots of the polynomial x n - p x n - 1 - q x - 1 for p ⩾ q ⩾ 1 . This is the characteristic polynomial of the recurrence relation F k , p , q ( n ) = p F k , p , q ( n - 1 ) + q F k , p , q ( n - k + 1 ) + F k , p , q ( n - k ) for n ⩾ k , which includes the relations of several particular sequences recently defined. In the end, a matricial representation for such a recurrence relation is provided.

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