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

PDF
 Accepted PostPrint Version
Download (1MB)  Preview 

PDF (Cardiff University Electronic Publication Form.)
 Supplemental Material
Restricted to Repository staff only Download (1MB) 
Abstract
For nonnegative integers r we examine four families of alternating and nonalternating 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Status:  Unpublished 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 
Actions (repository staff only)
Edit Item 