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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

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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 04, 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. 04 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 04]. 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