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Tulane University

1. Kesarwani, Aashita. Theory of the generalized modified Bessel function K_{z,w}(x) and 2-adic valuations of integer sequences.

Degree: 2017, Tulane University

URL: https://digitallibrary.tulane.edu/islandora/object/tulane:77514

Modular-type transformation formulas are the identities that are invariant under the transformation α → 1/α, and they can be represented as F (α) = F (β) where α β = 1. We derive a new transformation formula of the form F (α, z, w) = F (β, z, iw) that is a one-variable generalization of the well-known Ramanujan-Guinand identity of the form F (α, z) = F (β, z) and a two-variable generalization of Koshliakov’s formula of the form F (α) = F (β) where α β = 1. The formula is generated by first finding an integral J that is comprised of an invariance function Z and evaluating the integral to give F (α, z, w) mentioned above. The modified Bessel function K z (x) appearing in Ramanujan-Guinand identity is generalized to a new function, denoted as K z,w (x), that yields a pair of functions reciprocal in the Koshliakov kernel, which in turn yields the invariance function Z and hence the integral J and the new formula. The special function K z,w (x), first defined as the inverse Mellin transform of a product of two gamma functions and two confluent hypergeometric functions, is shown to exhibit a rich theory as evidenced by a number of integral and series representations as well as a differential-difference equation. The second topic of the thesis is 2-adic valuations of integer sequences associated with quadratic polynomials of the form x 2 +a. The sequence {n 2 +a : n ∈ Z} contains numbers divisible by any power of 2 if and only if a is of the form 4 m (8l+7). Applying this result to the sequences derived from the sums of four or fewer squares when one or more of the squares are kept constant leads to interesting results, that also points to an inherent connection with the functions r k (n) that count the number of ways to represent n as sums of k integer squares. Another class of sequences studied is the shifted sequences of the polygonal numbers given by the quadratic formula, for which the most common examples are the triangular numbers and the squares.

1

Aashita Kesarwani

Subjects/Keywords: Bessel functions; Theta transformation formula; Riemann zeta function

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Kesarwani, A. (2017). Theory of the generalized modified Bessel function K_{z,w}(x) and 2-adic valuations of integer sequences. (Thesis). Tulane University. Retrieved from https://digitallibrary.tulane.edu/islandora/object/tulane:77514

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Kesarwani, Aashita. “Theory of the generalized modified Bessel function K_{z,w}(x) and 2-adic valuations of integer sequences.” 2017. Thesis, Tulane University. Accessed August 13, 2020. https://digitallibrary.tulane.edu/islandora/object/tulane:77514.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Kesarwani, Aashita. “Theory of the generalized modified Bessel function K_{z,w}(x) and 2-adic valuations of integer sequences.” 2017. Web. 13 Aug 2020.

Vancouver:

Kesarwani A. Theory of the generalized modified Bessel function K_{z,w}(x) and 2-adic valuations of integer sequences. [Internet] [Thesis]. Tulane University; 2017. [cited 2020 Aug 13]. Available from: https://digitallibrary.tulane.edu/islandora/object/tulane:77514.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Kesarwani A. Theory of the generalized modified Bessel function K_{z,w}(x) and 2-adic valuations of integer sequences. [Thesis]. Tulane University; 2017. Available from: https://digitallibrary.tulane.edu/islandora/object/tulane:77514

Not specified: Masters Thesis or Doctoral Dissertation

Tulane University

2. Ngo, Tri. A rigorous proof of some heuristic rules in the Method of Brackets to evaluate definite integrals.

Degree: 2018, Tulane University

URL: https://digitallibrary.tulane.edu/islandora/object/tulane:87832

Many symbolic integration methods and algorithms have been developed to deal with definite integrals such as those of interest to physicists, theoretical chemists and engineers. These have been implemented into mathematical softwares like Maple and Mathematica to give closed forms of definite integrals. The work presented here introduces and analytically investigates an algorithm called Method of brackets. Method of brackets consists of a small number of rules to transform the evaluation of a definite integral into a problem of solving a system of linear equations. These rules are heuristic so justification is needed to make this method rigorous. Here we use contour integrals to justify the evaluation given by the algorithm.

1

Tri Ngo

Subjects/Keywords: Method of brackets

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Ngo, T. (2018). A rigorous proof of some heuristic rules in the Method of Brackets to evaluate definite integrals. (Thesis). Tulane University. Retrieved from https://digitallibrary.tulane.edu/islandora/object/tulane:87832

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Ngo, Tri. “A rigorous proof of some heuristic rules in the Method of Brackets to evaluate definite integrals.” 2018. Thesis, Tulane University. Accessed August 13, 2020. https://digitallibrary.tulane.edu/islandora/object/tulane:87832.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Ngo, Tri. “A rigorous proof of some heuristic rules in the Method of Brackets to evaluate definite integrals.” 2018. Web. 13 Aug 2020.

Vancouver:

Ngo T. A rigorous proof of some heuristic rules in the Method of Brackets to evaluate definite integrals. [Internet] [Thesis]. Tulane University; 2018. [cited 2020 Aug 13]. Available from: https://digitallibrary.tulane.edu/islandora/object/tulane:87832.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Ngo T. A rigorous proof of some heuristic rules in the Method of Brackets to evaluate definite integrals. [Thesis]. Tulane University; 2018. Available from: https://digitallibrary.tulane.edu/islandora/object/tulane:87832

Not specified: Masters Thesis or Doctoral Dissertation

Tulane University

3. Guan, Xiao. Methods in Symbolic Computation and p-adic Valuations of Polynomials.

Degree: 2017, Tulane University

URL: https://digitallibrary.tulane.edu/islandora/object/tulane:76381

1

Xiao Guan

Subjects/Keywords: elliptic integrals

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Guan, X. (2017). Methods in Symbolic Computation and p-adic Valuations of Polynomials. (Thesis). Tulane University. Retrieved from https://digitallibrary.tulane.edu/islandora/object/tulane:76381

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Guan, Xiao. “Methods in Symbolic Computation and p-adic Valuations of Polynomials.” 2017. Thesis, Tulane University. Accessed August 13, 2020. https://digitallibrary.tulane.edu/islandora/object/tulane:76381.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Guan, Xiao. “Methods in Symbolic Computation and p-adic Valuations of Polynomials.” 2017. Web. 13 Aug 2020.

Vancouver:

Guan X. Methods in Symbolic Computation and p-adic Valuations of Polynomials. [Internet] [Thesis]. Tulane University; 2017. [cited 2020 Aug 13]. Available from: https://digitallibrary.tulane.edu/islandora/object/tulane:76381.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Guan X. Methods in Symbolic Computation and p-adic Valuations of Polynomials. [Thesis]. Tulane University; 2017. Available from: https://digitallibrary.tulane.edu/islandora/object/tulane:76381

Not specified: Masters Thesis or Doctoral Dissertation