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Generating sequences from the sums of binomial coefficients in a residue class modulo q.

Humphreys, David M. 2020. Generating sequences from the sums of binomial coefficients in a residue class modulo q. PhD Thesis, Cardiff University.
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Abstract

For non-negative integers r we examine four families of alternating and non-alternating sign closed form binomial sums, Fs;ab(r; t; q), in a generalised congruence modulo q. We explore sums of squares and divisibility properties such as those determined by Weisman (and Fleck). Extending r to all integers we express the sequences in terms of closed form roots of unity and subsequently cosines. By a renumbering of these sequences we build eight new \diagonalised" sequences, Ls;abc(r; t; q), and construct equivalent closed forms and sums of squares relations. We modify Fibonacci type polynomials to construct order m recurrence polynomials that satisfy these diagonalised sequences. These recurrence polynomial sequences are shown to satisfy second order differential equations and exhibit orthogonal relations. From these latter relations we establish three term recurrence relations both between and within sequences. By the application of the reciprocal recurrence polynomial and hypergeometric functions, generating functions for these renumbered sequences are determined. Then employing these latter functions, we establish theorems that enable us to express each of the new sequences in terms of a Minor Corner Layered (MCL) determinant. When r is a negative integer and q = 2m+b is unspecified, the MCL determinants produce sequences of polynomials in m. For particular sequences we truncate these polynomials to contain only the leading coefficient and find that the truncated polynomial is equal to that of a Dirichlet series of the form zeta, lambda, beta or eta. From this relationship, recurrence polynomials for these latter functions are established Finally we develop a congruence for the denominator of the uncancelled modified Bernoulli numbers of the first kind, Bn=n!, and consequently a similar congruence for the zeta function at positive even valued integers. Furthermore we determine that these congruences obey the Fleck congruence.

Item Type: Thesis (PhD)
Date Type: Completion
Status: Unpublished
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Date of First Compliant Deposit: 18 November 2020
Last Modified: 20 Nov 2020 09:37
URI: http://orca.cf.ac.uk/id/eprint/136451

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