scholarly journals Polynomial series expansions for confluent and Gaussian hypergeometric functions

2005 ◽  
Vol 74 (252) ◽  
pp. 1937-1953 ◽  
Author(s):  
W. Luh ◽  
J. Müller ◽  
S. Ponnusamy ◽  
P. Vasundhra
Keyword(s):  
Hypergeometric Functions ◽  
Series Expansions ◽  
Gaussian Hypergeometric Functions ◽  
Polynomial Series

scholarly journals Coalescent lineage distributions

2006 ◽  
Vol 38 (02) ◽  
pp. 405-429 ◽  
Author(s):  
Robert C. Griffiths
Keyword(s):  
Brownian Motion ◽  
Hypergeometric Functions ◽  
Common Ancestor ◽  
Recent Common Ancestor ◽  
Series Expansions ◽  
Standard Brownian Motion ◽  
Most Recent Common Ancestor ◽  
Coalescent Tree ◽  
Expected Values ◽  
Special Case

We study identities for the distribution of the number of edges at time t back (i.e. measured backwards) in a coalescent tree whose subtrees have no mutations. This distribution is important in the infinitely-many-alleles model of mutation, where every mutation is unique. The model includes, as a special case, the number of edges in a coalescent tree at time t back when mutation is ignored. The identities take the form of expected values of functions of Z t =eiX t , where X t is distributed as standard Brownian motion. Associated identities are also found for the distributions of the time to the most recent common ancestor, the time until loss of ancestral lines by coalescence or mutation, and the age of a mutation. Hypergeometric functions play an important role in the identities. The identities are of mathematical interest, as well as potentially being formulae to use for numerical integration or simulation to compute distributions that are usually expressed as alternating-sign series expansions, which are difficult to compute.

scholarly journals Landen Inequalities for Zero-Balanced Hypergeometric Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Slavko Simić ◽  
Matti Vuorinen
Keyword(s):  
Open Problem ◽  
Hypergeometric Functions ◽  
Elliptic Integral ◽  
Complete Elliptic Integral ◽  
Turn On ◽  
Gaussian Hypergeometric Functions

For zero-balanced Gaussian hypergeometric functionsF(a,b;a+b;x),a,b>0, we determine maximal regions ofabplane where well-known Landen identities for the complete elliptic integral of the first kind turn on respective inequalities valid for eachx∈(0,1). Thereby an exhausting answer is given to the open problem from the work by Anderson et al., 1990.

scholarly journals On Mathieu-type series for the unified Gaussian hypergeometric functions

2020 ◽  
Vol 14 (1) ◽  
pp. 138-149
Author(s):  
Rakesh Parmar ◽  
Tibor Pogány
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Integral Form ◽  
Type A ◽  
Gauss Hypergeometric Function ◽  
Type Series ◽  
Gaussian Hypergeometric Functions ◽  
Special Cases

The main purpose of this paper is to present closed integral form expressions for the Mathieu-type a-series and for the associated alternating versions whose terms contain a generalized p-extended Gauss' hypergeometric function. Related bounding inequalities for the p-generalized Mathieu-type series are also obtained. Finally, a set of various (known or new) special cases and consequences of the results earned are presented.

scholarly journals Monotonicity, convexity properties and inequalities involving Gaussian hypergeometric functions with applications

2021 ◽  
Vol 7 (4) ◽  
pp. 4974-4991
Author(s):  
Ye-Cong Han ◽  
◽  
Chuan-Yu Cai ◽  
Ti-Ren Huang ◽  
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Elliptic Integral ◽  
Gaussian Hypergeometric Function ◽  
Gaussian Hypergeometric Functions ◽  
Convexity Properties

<abstract><p>In this paper, we mainly prove monotonicity and convexity properties of certain functions involving zero-balanced Gaussian hypergeometric function $ F(a, b; a+b; x) $. We generalize conclusions of elliptic integral to Gaussian hypergeometric function, and get some accurate inequalities about Gaussian hypergeometric function.</p></abstract>

scholarly journals Sufficiency for Gaussian hypergeometric functions to be uniformly convex

1999 ◽  
Vol 22 (4) ◽  
pp. 765-773 ◽  
Author(s):  
Yong Chan Kim ◽  
S. Ponnusamy
Keyword(s):  
Hypergeometric Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Hypergeometric Functions ◽  
Uniformly Convex ◽  
Gaussian Hypergeometric Functions

LetF(a,b;c;z)be the classical hypergeometric function andfbe a normalized analytic functions defined on the unit disk𝒰. Let an operatorIa,b;c(f)be defined by[Ia,b;c(f)](z)=zF(a,b;c;z)*f(z). In this paper the authors identify two subfamilies of analytic functionsℱ1andℱ2and obtain conditions on the parametersa,b,csuch thatf∈ℱ1impliesIa,b;c(f)∈ℱ2.

scholarly journals On Certain Sufficient Condition Involving Gaussian Hypergeometric Functions

2009 ◽  
Vol 2009 ◽  
pp. 1-15
Author(s):  
H. Silverman ◽  
Thomas Rosy ◽  
S. Kavitha
Keyword(s):  
Unit Disk ◽  
Hypergeometric Functions ◽  
Open Unit ◽  
Sufficient Condition ◽  
Open Unit Disk ◽  
New Class ◽  
Inclusion Properties ◽  
Complex Order ◽  
Gaussian Hypergeometric Functions

The authors define a new subclass of of functions involving complex order in the open unit disk . For this new class, we obtain certain inclusion properties involving the Gaussian hypergeometric functions.

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