ARTICLE | doi:10.20944/preprints201704.0040.v1
Subject: Mathematics & Computer Science, Analysis Keywords: Catalan number; integral representation; equivalent relation; application; sum of power series; Catalan--Qi function; Catalan--Qi number; beta function
Online: 7 April 2017 (04:27:13 CEST)
In the paper, the authors survey integral representations of the Catalan numbers and the Catalan–Qi function, discuss equivalent relations between these integral representations, supply alternative and new proofs of several integral representations, collect applications of some integral representations, and present sums of several power series whose coefficients involve the Catalan numbers.
ARTICLE | doi:10.20944/preprints201708.0056.v1
Subject: Mathematics & Computer Science, Computational Mathematics Keywords: expression; analytic property; Fuss–Catalan number; Catalan–Qi function; Catalan number; monotonicity; logarithmic convexity; complete monotonicity; minimality; inequality
Online: 15 August 2017 (08:36:23 CEST)
In the paper, the authors express the Fuss–Catalan numbers as several forms in terms of the Catalan–Qi function, find some analytic properties, including the monotonicity, logarithmic convexity, complete monotonicity, and minimality, of the Fuss–Catalan numbers, and derive a double inequality for bounding the Fuss–Catalan numbers.
ARTICLE | doi:10.20944/preprints201804.0305.v1
Subject: Mathematics & Computer Science, General Mathematics Keywords: Catalan beta function; Riemann’s zeta function; primes; Dirichlet L-function
Online: 24 April 2018 (05:10:20 CEST)
It is well known that the primes and prime powers have a deep relationship with the nontrivial zeros of Riemann’s zeta function. This is a reciprocal relationship. The zeros and the primes are encoded in each other and are reciprocally recoverable. Riemann’s zeta is an extended or continued version of Euler’s zeta function which in turn equates with Euler’s product formula over the primes. This paper shows that the zeros of the converging Dirichlet or Catalan beta function, which requires no continuation to be valid in the critical strip, can be easily determined. The imaginary parts of these zeros have a deep and reciprocal relationship with the odd primes and odd prime powers. This relationship separates the odd primes into those having either 1 or 3 as a remainder after division by 4. The vector pathway of the beta function is such that the real part of its zeros has to be a half.
ARTICLE | doi:10.20944/preprints201703.0029.v1
Subject: Mathematics & Computer Science, Analysis Keywords: series identity; Catalan number; Catalan function; Riemanian zeta function; alternative Hurwitz zeta function; digamma function
Online: 6 March 2017 (07:01:20 CET)
In the paper, the authors discover several series identities involving the Catalan numbers, the Catalan function, the Riemanian zeta function, and the alternative Hurwitz zeta function.
ARTICLE | doi:10.20944/preprints201711.0120.v1
Subject: Mathematics & Computer Science, Computational Mathematics Keywords: simplification; coefficient; nonlinear ordinary differential equation; generating function; Catalan number; inverse matrix; lower triangular integer matrix; Faά di Bruno formula; Bell polynomial of the second kind; inversion theorem
Online: 20 November 2017 (07:20:26 CET)
In the paper, by the Faά di Bruno formula, several identities for the Bell polynomials of the second kind, and an inversion theorem, the authors simplify coefficients of two families of nonlinear ordinary differential equations for the generating function of the Catalan numbers and discover inverses of fifteen closely related lower triangular integer matrices.
ARTICLE | doi:10.20944/preprints201610.0089.v1
Subject: Mathematics & Computer Science, Analysis Keywords: improper integral; explicit expression; unified expression; beta function; Wallis ratio; integral representation; Catalan number
Online: 21 October 2016 (08:29:08 CEST)
In the paper, the author presents explicit and unified expressions for a sequence of improper integrals in terms of the beta functions and the Wallis ratios. Hereafter, the author derives integral representations for the Catalan numbers originating from combinatorics.
ARTICLE | doi:10.20944/preprints201703.0200.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: explicit formula; recursive formula; generalized Motzkin number; Motzkin number; restricted hexagonal number; Catalan number; generating function
Online: 27 March 2017 (11:02:41 CEST)
In the paper, the authors find two explicit formulas and recover a recursive formula for the generalized Motzkin numbers. Consequently, the authors deduce two explicit formulas and a recursive formula for the Motzkin numbers, the Catalan numbers, and the restricted hexagonal numbers respectively.
ARTICLE | doi:10.20944/preprints201703.0209.v2
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: identity; inverse matrix; explicit formula; generating function; Chebyshev polynomials of the second kind; Catalan number; triangular matrix; classical hypergeometric function; integral representation
Online: 7 August 2017 (15:49:11 CEST)
In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers respectively with the Chebyshev polynomials, the central Delannoy numbers, and the Fibonacci polynomials.
ARTICLE | doi:10.20944/preprints201703.0119.v1
Subject: Mathematics & Computer Science, Analysis Keywords: Lévy–Khintchine representation; integral representation; Bernstein function; Stieltjes function; Toader–Qi mean; weighted geometric mean; Bessel function of the first kind; probabilistic interpretation; probabilistic interpretation; application in engineering; inequality
Online: 16 March 2017 (11:31:31 CET)
In the paper, by virtue of a Lévy–Khintchine representation and an alternative integral representation for the weighted geometric mean, the authors establish a Lévy–Khintchine representation and an alternative integral representation for the Toader–Qi mean. Moreover, the authors also collect an probabilistic interpretation and applications in engineering of the Toader–Qi mean.
HYPOTHESIS | doi:10.20944/preprints202104.0760.v1
Subject: Medicine & Pharmacology, Allergology Keywords: Complementary and alternative medicine; Traditional Chinese medicine (TCM); Thermodynamic mechanism; Entropic systems biology; Self-organized criticality (SOC); Self-organization triggering factor (SOTF); Wuxing (five phases), Qi, Meridians (Jingluo); Acupuncture points; Chinese herbs; Aquamoleculomics; Modernization of TCM
Online: 28 April 2021 (17:12:29 CEST)
As a complementary and alternative medicine in the western countries for decades, traditional Chinese medicine (TCM) has been used for more than 2000 years in China. Because of the characteristics of the philosophical style and the unknown mechanism of action, TCM sometimes has been biasedly described as "fraught with pseudoscience". From the scientific basis of the systems biology, here we promoted a novel medical model called the entropic systems medicine which could be applied to scientize TCM in future. In entropic systems medicine, TCM and Western modern biomedicine target the different variables of the entropic system. For instance, while Western modern biomedicine directly targets the phenotypes and its SOCs of macrostates, TCM differently targets the microstates, entropy and entropic force to generate SOTFs gradually causing the differentiated syndromes to be slowly rearranged. The prerequisites to modernize TCM are the entropic systems biology having been well established so that the variables could be precisely monitored and mathematically calculated.