Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

Enumeration of permutations by number of alternating runs

Andrews–Gordon type series for Capparelli's and Göllnitz–Gordon identities

We prove seven of the Rogers-Ramanujan type identities modulo $12$ that were conjectured by Kanade and Russell. Included among these seven are the two original modulo $12$ identities, in which the products have asymmetric congruence conditions, as well as the three symmetric identities related to the principally specialized characters of certain level $2$ modules of $A_9^{(2)}$. We also give reductions of four other conjectures in terms of single-sum basic hypergeometric series.

Proofs and reductions of various conjectured partition identities of Kanade and Russell

. Let Bk,i(n) be the number of partitions of n with certain difference condition and let Ak,i(n) be the number of partitions of n with certain congruence condition. The RogersRamanujan-Gordon theorem states that Bk,i(n) = Ak,i(n). Lovejoy obtained an overpartition analogue of the Rogers-Ramanujan-Gordon theorem for the cases i = 1 and i = k. We find an overpartition analogue of the Rogers-Ramanujan-Gordon theorem in the general case. Let Dk,i(n) be the number of overpartitions of n satisfying certain difference condition and Ck,i(n) be the number of overpartitions of n whose non-overlined parts satisfy certain congruences condition. We show that Ck,i(n) = Dk,i(n). By using a function introduced by Andrews, we obtain a recurrence relation which implies that the generating function of Dk,i(n) equals the generating function of Ck,i(n). We also find a generating function formula of Dk,i(n) by using Gordon marking representations of overpartitions, which can be considered as an overpartition analogue of an identity of Andrews for ordinary partitions.

The Rogers–Ramanujan–Gordon theorem for overpartitions

In 1974, Andrews discovered the generating function for the partitions of n considered in a theorem due to Gordon. In a more recent paper, he reconsidered this generating function and gave refinements where additional restrictions involving parities are imposed. A combinatorial construction for the partitions enumerated by the mentioned generating function is given. Some of the Andrews' refinements are proven combinatorially, and a conjecture of his is settled.

Parity considerations in Andrews-Gordon identities

We construct Andrews-Gordon type evidently positive series as generating functions for the partitions satisfying the difference conditions imposed by Capparelli's identities and Gollnitz-Gordon identities. The construction involves building base partitions, and move parts around subject to certain rules. Some missing cases are also considered, but no new identities are given.

Andrews-Gordon Type Series for Capparelli's and G\"{o}llnitz-Gordon Identities

We construct Andrews-Gordon type evidently positive series as generating functions of partitions satisfying certain difference conditions in six conjectures by Kanade and Russell. We construct generating functions for missing partition enumerants, naturally without claiming new partition identities. Thus, we obtain $q$-series conjectures as companions to Kanade and Russell's combinatorial conjectures.

Andrews-Gordon Type Series for Kanade-Russell Conjectures

https://etda.libraries.psu.edu/files/final_submissions/3571

Kağan Kurşungöz

Papers

Parity considerations in Andrews-Gordon identities

Andrews–Gordon type series for Capparelli's and Göllnitz–Gordon identities

Andrews-Gordon Type Series for Capparelli's and G\"{o}llnitz-Gordon Identities

Andrews-Gordon Type Series for Kanade-Russell Conjectures

PARITY CONSIDERATIONS IN ANDREWS-GORDON IDENTITIES, AND THE k-MARKED DURFEE SYMBOLS