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The spt-crank for overpartitions

Frank G. Garvan, Chris Jennings-Shaffer (2014)

Acta Arithmetica

Bringmann, Lovejoy, and Osburn (2009, 2010) showed that the generating functions of the spt-overpartition functions s p t ¯ ( n ) , s p t ¯ 1 ( n ) , s p t ¯ 2 ( n ) , and M2spt(n) are quasimock theta functions, and satisfy a number of simple Ramanujan-like congruences. Andrews, Garvan, and Liang (2012) defined an spt-crank in terms of weighted vector partitions which combinatorially explain simple congruences modulo 5 and 7 for spt(n). Chen, Ji, and Zang (2013) were able to define this spt-crank in terms of ordinary partitions. In this...

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.

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