Introduction To Hypergeometric, Supertrigonometric, And Superhyperbolic Functions

504 Pages·2021·3.088 MB·English

by  Xiao-Jun Yang

An Introduction to
Hypergeometric,
Supertrigonometric, and
Superhyperbolic Functions

An Introduction to
Hypergeometric,
Supertrigonometric, and
Superhyperbolic Functions
Xiao-Jun Yang
StateKeyLaboratoryforGeomechanicsandDeepUndergroundEngineering
SchoolofMathematics
ChinaUniversityofMiningandTechnology
Xuzhou,Jiangsu,China
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Contents
Biography xi
Preface xiii
1. Euler gamma function, Pochhammer symbols and Euler beta
function 1
1.1. Eulergammafunction 1
1.2. Pochhammersymbols 6
1.3. Eulerbetafunction 9
2. Hypergeometric, supertrigonometric, and superhyperbolic
functions via Clausen hypergeometric series 13
2.1. Clausenhypergeometricseries 13
2.2. The hypergeometric supertrigonometric functions via
Clausenhypergeometricseries 49
2.3. ThehypergeometricsuperhyperbolicfunctionsviaClausen
hypergeometricseries 73
2.4. The special functions via Clausen hypergeometric series
with three numerator parameters and two denominator
parameters 97
2.5. Analytic number theory via Clausen hypergeometric
functions 115
3. Hypergeometric supertrigonometric and superhyperbolic
functions via Gauss hypergeometric series 139
3.1. Gausshypergeometricseries 139
3.2. Hypergeometric supertrigonometric functions via Gauss
hypergeometricseries 145
3.3. Hypergeometric superhyperbolic functions via Gauss
hypergeometricseries 149
vii
viii Contents
3.4. SomeelementaryexamplesfortheGausshypergeometric
series 154
3.5. Integral representations for the hypergeometric
superhyperbolic and hypergeometric superhyperbolic
functions 169
3.6. AnalyticnumbertheoryviaGausshypergeometricfunctions 203
4. Hypergeometric supertrigonometric and superhyperbolic
functions via Kummer confluent hypergeometric series 237
4.1. TheKummerconfluenthypergeometricseriesoffirsttype 237
4.2. The hypergeometric supertrigonometric functions via
Kummerconfluenthypergeometricseriesoffirsttype 242
4.3. ThehypergeometricsuperhyperbolicfunctionsviaKummer
confluenthypergeometricseriesoffirsttype 254
4.4. TheKummerconfluenthypergeometricseriesofsecondtype 265
4.5. The hypergeometric supertrigonometric functions via
Kummerconfluenthypergeometricseriesofsecondtype 267
4.6. ThehypergeometricsuperhyperbolicfunctionsviaKummer
confluenthypergeometricseriesofsecondtype 276
4.7. Analytic number theory via Kummer confluent
hypergeometricseries 284
5. Hypergeometric supertrigonometric and superhyperbolic
functions via Jacobi polynomials 293
5.1. Jacobipolynomials 293
5.2. Jacobi–Lukepolynomials 309
5.3. Jacobi–Luke-typepolynomials 312
6. Hypergeometric supertrigonometric and superhyperbolic
functions via Laguerre polynomials 345
6.1. Laguerrepolynomials 345
6.2. ExtendedworkscontainingtheLaguerrepolynomials 390
6.3. Someresultsbasedonthespecialfunctions 417
Contents ix
7. Hypergeometric supertrigonometric and superhyperbolic
functions via Legendre polynomials 445
7.1. Legendrepolynomials 445
7.2. Legendre-typepolynomials 456
References 471
Index 481