Distance-Regular Graphs

1. The real variable 2. Scalars and vectors 3. Tensors 4. Matrices 5. Multiple integrals 6. Potential theory 7. Operational methods 8. Physical applications of the operational method 9. Numerical methods 10. Calculus of variations 11. Functions of a complex variable 12. Contour integration and Bromwich's integral 13. Contour integration 14. Fourier's theorem 15. The factorial and related functions 16. Solution of linear differential equations of the second order 17. Asymptotic expansions 18. The equations of potential, waves and heat conduction 19. Waves in one dimension and waves with spherical symmetry 20. Conduction of heat in one and three dimensions 21. Bessel functions 22. Applications of Bessel functions 23. The confluent hypergeometric function 24. Legendre functions and associated functions 25. Elliptic functions Notes Appendix on notation Index.

Methods of Mathematical Physics. By Harold Jeffreys and Bertha Swirles Jeffreys. Pp. viii, 679. 63s. 1946. (Cambridge University Press)

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Introduction To Cyclotomic Fields

Introduction to coding theory (2nd edition), by J.H. van Lint. Pp.183 DM86. 1992. ISBN 0-387-54894-7 (Springer)

The first goal of algebraic number theory is the generalization of the theorem on the unique representation of natural numbers as products of prime numbers to algebraic numbers. Gauss considered the ring \( \mathbb{Z}\left[ {\sqrt {{ - 1}} } \right] \) of all numbers of the form \( a + \sqrt {{ - 1b}} \) with \( a,\;b \in \mathbb{Z} \) and showed that \( \mathbb{Z}\left[ {\sqrt {{ - 1}} } \right] \) is a ring with unique factorization in prime elements (see §2.1). He introduced these numbers for the development of his theory of biquadratic residues. Another motivation for the study of the arithmetic of algebraic numbers comes from the theory of Tiophantine equations. For example, the quadratic form \( f({x_1},\;{x_2}) = x_1^2 - Dx_2^2 \) with \( D \in \mathbb{Z} \), \( \sqrt {{D\; \notin \;\mathbb{Z}}} \) can be written in the form \( \left( {{x_1} - \sqrt {{D{x_2}}} } \right)\left( {{x_1} + \sqrt {{D{x_2}}} } \right) \). Hence the question about the representation of integers by f(a 1, a 2) with \( {a_1},\;{a_2} \in \mathbb{Z} \) can be reformulated as a question of factorization of algebraic numbers of the form \( {a_1} + \sqrt {{D{a_2}}} \). These numbers form a module in the field \( \mathbb{Q}(\sqrt {D} ) \).

Basic Number Theory

Studying degenerate versions of various special polynomials has became an active area of research and has yielded many interesting arithmetic and combinatorial results. Here we introduce a degenerate version of the polylogarithm function, the so-called degenerate polylogarithm function. Then we construct a new type of degenerate Bernoulli polynomial and number, the so-called degenerate poly-Bernoulli polynomial and number, by using the degenerate polylogarithm function, and derive several properties concerning the degenerate poly-Bernoulli numbers.

A Note on a New Type of Degenerate Bernoulli Numbers

In this paper, we introduce the degenerate Laplace transform and degenerate gamma function and investigate some properties of the degenerate Laplace transform and degenerate gamma function. From our investigation, we derive some interesting formulas related to the degenerate Laplace transform and degenerate gamma function.

Degenerate Laplace transform and degenerate gamma function

We consider the Witt-type formula for Daehee numbrers and polynomials and investigate some properties of those numbers and polynomials. In particular, Daehee numbers are closely related to higher-order Bernoulli numbers and Bernoulli numbers of the second kind.

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Daehee numbers and polynomials

In this paper, we introduce the degenerate Laplace transform and degenerate gamma function and investigate some of their properties. From our investigation, we derive some interesting formulas related to the degenerate Laplace transform and degenerate gamma function.

In this paper, we introduce polyexponential functions as an inverse to the polylogarithm functions, construct type 2 poly-Bernoulli polynomials by using this and derive various properties of type 2 poly-Bernoulli numbers. Then we introduce unipoly functions attached to each suitable arithmetic function as a universal concept which includes the polylogarithm and polyexponential functions as special cases. By making use of unipoly functions, we define unipoly-Bernoulli polynomials, type 2 unipoly-Bernoulli numbers, and unipoly-Bernoulli numbers of the second kind, and show some basic properties for them.

Dae San Kim

Papers

A Note on a New Type of Degenerate Bernoulli Numbers

Degenerate Laplace transform and degenerate gamma function

Daehee numbers and polynomials

Degenerate Laplace transform and degenerate gamma function

A Note on Polyexponential and Unipoly Functions