scholarly journals On Binomial Coefficient Residues

1957 ◽  
Vol 9 ◽  
pp. 363-370 ◽  
Author(s):  
J . B. Roberts
Keyword(s):  
Prime Number ◽  
Difference Equations ◽  
Binomial Coefficient ◽  
Binomial Coefficients ◽  
Linear Difference Equations ◽  
Constant Coefficients

The number of binomial coefficients , which are congruent to j , 0 ≤ j ≤ p − 1, modulo the prime number p is denoted by θj(n). In this paper we give systems of simultaneous linear difference equations with constant coefficients whose solutions would yield the quantities θj(n) explicitly.

scholarly journals Note on some representations of general solutions to homogeneous linear difference equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Stevo Stević ◽  
Bratislav Iričanin ◽  
Witold Kosmala ◽  
Zdeněk Šmarda
Keyword(s):  
General Solution ◽  
Difference Equation ◽  
Difference Equations ◽  
Fibonacci Sequence ◽  
Linear Difference Equation ◽  
Linear Difference Equations ◽  
Second Order Difference Equation ◽  
Constant Coefficients ◽  
Order Difference Equation ◽  
Homogeneous Linear

Abstract It is known that every solution to the second-order difference equation $x_{n}=x_{n-1}+x_{n-2}=0$ x n = x n − 1 + x n − 2 = 0 , $n\ge 2$ n ≥ 2 , can be written in the following form $x_{n}=x_{0}f_{n-1}+x_{1}f_{n}$ x n = x 0 f n − 1 + x 1 f n , where $f_{n}$ f n is the Fibonacci sequence. Here we find all the homogeneous linear difference equations with constant coefficients of any order whose general solution have a representation of a related form. We also present an interesting elementary procedure for finding a representation of general solution to any homogeneous linear difference equation with constant coefficients in terms of the coefficients of the equation, initial values, and an extension of the Fibonacci sequence. This is done for the case when all the roots of the characteristic polynomial associated with the equation are mutually different, and then it is shown that such obtained representation also holds in other cases. It is also shown that during application of the procedure the extension of the Fibonacci sequence appears naturally.

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