All a b c d e f g h i j k l m n o p q r s t u v w x y z Other

Previous Page 17 Next

Displaying 321 – 340 of 351

Showing per page

The tangent function and power residues modulo primes

Zhi-Wei Sun (2023)

Czechoslovak Mathematical Journal

Let p be an odd prime, and let a be an integer not divisible by p . When m is a positive integer with p ≡ 1 ( mod 2 m ) and 2 is an m th power residue modulo p , we determine the value of the product ∏ k ∈ R m ( p ) ( 1 + tan ( π a k / p ) ) , where R m ( p ) = { 0 < k < p : k ∈ ℤ is an m th power residue modulo p } . In particular, if p = x 2 + 64 y 2 with x , y ∈ ℤ , then ∏ k ∈ R 4 ( p ) 1 + tan π a k p = ( - 1 ) y ( - 2 ) ( p - 1 ) / 8 .

The two-parameter class of Schröder inversions

J. Schröder (2013)

Commentationes Mathematicae Universitatis Carolinae

Infinite lower triangular matrices of generalized Schröder numbers are used to construct a two-parameter class of invertible sequence transformations. Their inverses are given by triangular matrices of coordination numbers. The two-parameter class of Schröder transformations is merged into a one-parameter class of stretched Riordan arrays, the left-inverses of which consist of matrices of crystal ball numbers. Schröder and inverse Schröder transforms of important sequences are calculated.

Currently displaying 321 – 340 of 351