A Comprehensive Treatment of q-calculus
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1 A Comprehensive Treatment of q-calculus Bearbeitet von Thomas Ernst 1. Auflage Buch. xvi, 492 S. Hardcover ISBN Format (B x L): 15,5 x 23,5 cm Gewicht: 919 g Weitere Fachgebiete > Mathematik > Mathematische Analysis > Differentialrechnungen und -gleichungen Zu Leseprobe schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, ebooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte.
2 Contents 1 Introduction A survey of the chapters What is q-calculus? Elementaryseriesmanipulations Update on q-calculus Current textbooks on this subject Comparison with complex analysis Comparison with nonstandard analysis Comparisonwiththeunitsofphysics Analogies between q-analysis and analysis The first q-functions The different languages of q-calculus Schools traditions Ramifications and minor Schools Differentnotations Finite differences and Bernoulli numbers Umbral calculus, interpolation theory Elliptic and Theta Schools and notations, the oldest roots the q-forerunners Trigonometry, prosthaphaeresis, logarithms Thedevelopmentofcalculus The Faulhaber mathematics Descartes, Leibniz, Hindenburg, Arbogast TheFakultäten Königsberg School Viennese School Göttingen School The combinatorial School: Gudermann, Grünert Heidelberg School Weierstraß, formal power series and the Ɣ function xi
3 xii Contents 2.17 Halle q-analysis School Jakob Friedrich Fries, Martin Ohm, Babbage, Peacock and Herschel Different styles in q-analysis Pre q-analysis The early connection between analytic number theory and q-series Some aspects of combinatorical identities FaàdiBrunoformula The duality between Bernoulli and Stirling numbers Tangent numbers, Euler numbers The occurrence of binomial coefficient identities in the literature Nineteenth century: Catalan, Grigoriew, Imchenetsky A short history of hypergeometric series The Ɣ function Balanced and well-poised hypergeometric series Fractional differentiation Newton,Taylor,Stirling,Montmort Euler s contribution Vandermonde and Pfaffian summation formulas Conic sections in the seventeenth century TheinfinityinEngland The infinity in the hands of Euler The infinity, the binomial coefficients Gauß contribution After Gauß; Clausen, Jacobi Kummer s contribution Cauchy, Riemann, Heine, Thomae, Papperitz ; Sonine, Goursat, Stieltjes, Schafheitlin, Pochhammer, Mellin First half of the twentieth century; England, USA Special functions defined by integrals Second half of the twentieth century The Jacobi theta functions; different notations; properties Meromorphic continuation and Riemann surfaces Wave equation Orthogonal polynomials Legendre-d Allonville-Murphy polynomials Laguerre-Abel-Sonine-Murphy-Chebyshev-Halphen- Szegő polynomials Jacobi polynomials Hermite polynomials The q-umbral calculus and semigroups. The Nørlund calculus of finite differences The q-umbral calculus and semigroups Finite differences...104
4 Contents xiii 4.3 q-appell polynomials The generalized q-bernoulli polynomials The Ward q-bernoullinumbers The generalized JHC q-bernoulli polynomials NWA q-euler polynomials The NWA generalized q-eulernumbers Several variables; n negative q-euler-maclaurin expansions JHC polynomials of many variables; negative order JHC q-euler-maclaurin expansions Eulersymbolicformula Complementary argument theorems q-lucas and q-g polynomials q-lucas numbers The q-g polynomials Lucas and G polynomials of negative order Expansion formulas ThesemiringofWardnumbers q-stirling numbers Introduction The Hahn-Cigler-Carlitz-Johnson approach The Carlitz-Gould approach The Jackson q-derivative as difference operator Applications The first q-functions q-analogue, q-factorial, tilde operator The q-derivative The q-integral Two other tilde operators The Gaussian q-binomial coefficients and the q-leibniz theorem Otherformulas Cigler s operational method for q-identities Gould and Carlitz q-binomial coefficient identities q-exponential and q-trigonometric functions TheHeinefunction Oscillations in q-analysis The Jackson-Hahn-Cigler q-addition and q-analogues of the trigonometric functions Further q-trigonometric functions The Nalli-Ward-Al-Salam q-addition and some variants of the q-difference operator Weierstraß elliptic functions and sigma functions Elliptic functions Connections with the Ɣ q function The Chen-Liu operator or parameter augmentation...239
5 xiv Contents 7 q-hypergeometric series Definition of q-hypergeometric series q-difference equation for p+1 φ p Balanced and well-poised q-hypergeometric series Advantages of the Heine definition q-binomial theorem Jacobi s elliptic functions expressed as real and imaginary parts of q-hypergeometric series with exponential argument (Heine) The Jacobi triple product identity q-contiguityrelations Heine q-transformations The q-betafunction Heines q-analogue of the Gauß summation formula A q-analogue of the Pfaff-Saalschütz summation formula Sears 4 φ 3 transformation q-analogues of Thomae s transformations TheBailey-Daumsummationformula A general expansion formula A summation formula for a terminating very-well-poised 4 φ 3 series A summation formula for a terminating very-well-poised 6 φ 5 series Watson s transformation formula for a terminating very-wellpoised 8 φ 7 series Jackson s sum of a terminating very-well-poised balanced 8φ 7 series Threecorollaries Watson s proof of the Rogers-Ramanujan identities Bailey s 1929 transformation formula for a terminating, balanced, very-well-poised 10 φ Watson s q-analogue of the Barnes contour integral Three q-analogues of the Euler integral formula for the function Ɣ(x) Inequalities for the Ɣ q function Summaryoftheumbralmethod Sundry topics Four q-summationformulasofandrews Some quadratic q-hypergeometric transformations The Kummer 2 F 1 ( 1) formula and Jacobi s theta function Another proof of the q-dixonformula A finite version of the q-dixonformula The Jackson summation formula for a finite, 2-balanced, well-poised 5 φ 4 series The Jackson finite q-analogue of the Dixon formula Other examples of q-special functions q-analogues of two formulas by Brown and Eastham...296
6 Contents xv 8.10 The q-analogue of Truesdell s function The Bailey transformation for q-series q-taylor formulas with remainder; the mean value theorem The mean value theorem in q-analysis Bilateralseries Fractional q-integrals q-orthogonal polynomials Ciglerian q-laguerre polynomials The different Laguerre-philosophies The q-laguerre polynomials Generating functions and recurrences Product expansions Bilinear generating functions Al-Salam operator expressions The q-laguerre Rodriguez operator q-orthogonality q-jacobi polynomials Definition and the Rodriguez formula The q-jacobi Rodriguez operator More generating functions and recurrences q-orthogonality q-legendre polynomials and Carlitz-Al-Salam polynomials q-legendre polynomials Carlitz-Al-Salam polynomials q-functions of several variables The corresponding vector notation Historical introduction Transformations for basic double series Double q-balanced series Transformation formula of Carlitz-Srivastava ThreeformulasofAndrews q-analogues of Carlson s formulas The q-appell function 1 as q-integral q-analogues of some of Srivastava s formulas Some q-formulasofsrivastava Generating functions Transformations Double sum identities (Srivastava and Jain) Two reduction formulas of Karlsson and Srivastava q-analogues of reducibility theorems of Karlsson q-analogues of Burchnall-Chaundy expansions q-analogues of Verma expansions Asimilarformula Multiple extensions of the Rothe-von Grüson-Gauß formula
7 xvi Contents An expansion formula in the spirit of Chaundy Formulas according to Burchnall-Chaundy and Jackson Linear partial q-difference equations Introduction Canonical equations and symmetry techniques for q-series (Kalnins, Miller) q-difference equations for q-appell and q-lauricella functions q-calculus and physics The q-coulomb problem and the q-hydrogen atom Connections to knot theory General relativity Molecular and nuclear spectroscopy Elementary particle physics and chemical physics Electroweakinteraction String theory Wess-Zumino model Quantum Chromodynamics References Index before Index after Physics index Name index before Name index after Name index Physics Notation index Chapter 1, 2, Notation index Chapter Notation index Chapter 4, Notation index Chapter Notation index Chapter
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