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  • 1.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. MATHEMATICS I-V.
    A method for q-calculus2003Ingår i: J. Nonlinear Math. Physics, Vol. 10, nr 4, s. 487-525Artikel i tidskrift (Refereegranskat)
  • 2.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    A new method and its application to generalized q-Bessel polynomials2001Rapport (Övrigt vetenskapligt)
  • 3.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    A new method for q-hypergeometric series.2001Ingår i: Czech. J. of Physics, 2001, Vol. 51, nr 12Konferensbidrag (Övrigt vetenskapligt)
  • 4.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    A new method q-calculus2002Doktorsavhandling, monografi (Övrigt vetenskapligt)
    Abstract [en]

    In q-calculus we are looking for q-analogues of mathematical objects, which have the original object as limits when q tends to 1. q-Calculus has wide-ranging applications in analytic number theory and theoretical physics. The main topic of the thesis is the invention of the tilde operator and the renaissance of the q-addition. There are two types of q-addition, the Ward-AlSalam q-addition and the Hahn q-addition. The first is both commutative and associative, while the second is neither. This is one of the reasons why sometimes more than one q-analogue exist. These two operators form the basis of the method which unites hypergeometric series and q-hypergeometric series and which gives many formulas of q-calculus a natural form reminding directly of their classical origin. This method is reminiscent of Heine, who mentioned the case where one parameter in a q-hypergeometric series is plus infinity. The q-addition is the natural way to extend addition to the q-case as is shown when restating addition formulas for q-trigonometric functions.

    We give a more lucid definition of the q-difference operator. A new notation for powers of q reminding of the exponential function is given. A q-Taylor formula with remainder term expressed as q-integral is proved.

    We present a new expression for generalized Vandermonde determinants, and thus for the Schur function. We also obtain an equivalence relation on the set of all generalized Vandermonde determinants. We find a more general expression for the Vandermonde determinant. We show the connection to a determinant of Flowe and Harris and to the solution of difference and q-difference equations with constant coefficients.

  • 5.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    A new Vandermonde-related determinant and its connection to difference equations2000Rapport (Övrigt vetenskapligt)
  • 6.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    An umbral approach to find q-analogues of matrix formulas2013Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 439, nr 4, s. 1167-1182Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A general introduction is given to the logarithmic q-analogue formulation of mathematical expressions with a special focus on its use for matrix calculations. The fundamental definitions relevant to q-analogues of mathematical objects are given and form the basis for matrix formulations in the paper. The umbral approach is used to find q-analogues of significant matrices. Finally, as an explicit example, a new formula for q-Cauchy-Vandermonde determinant containing matrix elements equal to q-numbers introduced by Ward is proved by using a new type of q-Stirling numbers together with Lagrange interpolation in Z(q).

  • 7.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Expansion formulas for Apostol type Q-Appell polynomials, and their special cases2018Ingår i: Le Matematiche, ISSN 2037-5298, E-ISSN 0373-3505, Vol. 73, nr 1, s. 3-24Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present identities of various kinds for generalized q-ApostolBernoulli and Apostol-Euler polynomials and power sums, which resemble q-analogues of formulas from the 2009 paper by Liu and Wang. These formulas are divided into two types: formulas with only q-ApostolBernoulli, and only q-Apostol-Euler polynomials, or so-called mixed formulas, which contain polynomials of both kinds. This can be seen as a logical consequence of the fact that the q-Appell polynomials form a commutative ring. The functional equations for Ward numbers operating on the q-exponential function, as well as symmetry arguments, are essential for many of the proofs. We conclude by finding multiplication formulas for two q-Appell polynomials of general form. This brings us to the q-H polynomials, which were discussed in a previous paper.

  • 8.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Generalized Vandermonde determinants2000Rapport (Övrigt vetenskapligt)
  • 9.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    ON EULERIAN q-INTEGRALS FOR SINGLE AND MULTIPLE q-HYPERGEOMETRIC SERIES2018Ingår i: COMMUNICATIONS OF THE KOREAN MATHEMATICAL SOCIETY, ISSN 1225-1763, Vol. 33, nr 1, s. 179-196Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we extend the two q-additions with powers in the umbrae, define a q-multinomial-coefficient, which implies a vector version of the q-binomial theorem, and an arbitrary complex power of a JHC power series is shown to be equivalent to a special case of the first q-Lauricella function. We then present several q-analogues of hypergeo-metric integral formulas from the two books by Exton and the paper by Choi and Rathie. We also find multiple q-analogues of hypergeometric integral formulas from the recent paper by Kim. Finally, we prove several multiple q-hypergeometric integral formulas emanating from a paper by Koschmieder, which are special cases of more general formulas by Exton.

  • 10.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    On several q-special matrices, including the q-Bernoulli and q-Euler matrices2018Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 542, s. 422-440Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In the spirit of our earlier articles [12,14,11], and our work [13], we define two dual q-Bernoulli polynomials, with corresponding vector and matrix forms. Following Aceto Trigiante [1], the q-L matrix, the indefinite q-integral of the q-Pascal matrix is the link between the q-Cauchy and the q-Bernoulli matrix. The q-analogue of the Bernoulli complementary argument theorem can be expressed in matrix form through the diagonal An matrix. For the q-Euler polynomials corresponding results are obtained. The umbral calculus for generating functions of q-Appell polynomials is shown to be equivalent to a transform method, which maps polynomials to matrices, a true q-analogue of Arponen [6]. This is manifested by the Vein [21] matrix, which occurs as the transform of the q-difference operator. The Aceto Trigiante shifted q-Bernoulli matrix has a simple connection to the q-Bernoulli Arponen matrix through the q-Pascal matrix. We reintroduce certain q-Stirling numbers is an element of 7L(q) from [12], which will be needed for the polynomial matrix definitions.

  • 11.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    On the q-exponential of matrix q-Lie algebras2017Ingår i: Special Matrices, ISSN 2300-7451, Vol. 5, nr 1, s. 36-50Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper, we define several new concepts in the borderline between linear algebra, Lie groups and q-calculus. We first introduce the ring epimorphism tau, the set of all inversions of the basis q, and then the important q-determinant and corresponding q-scalar products from an earlier paper. Then we discuss matrix q-Lie algebras with a modified q-addition, and compute the matrix q-exponential to form the corresponding n x n matrix, a so-called q-Lie group, or manifold, usually with q-determinant 1. The corresponding matrix multiplication is twisted under tau, which makes it possible to draw diagrams similar to Lie group theory for the q-exponential, or the so-called q-morphism. There is no definition of letter multiplication in a general alphabet, but in this article we introduce new q-number systems, the biring of q-integers, and the extended q-rational numbers. Furthermore, we provide examples of matrices in su(q)(4), and its corresponding q-Lie group. We conclude with an example of system of equations with Ward number coeficients.

  • 12.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    On the q-Lie group of q-Appell polynomial matrices and related factorizations2018Ingår i: Special Matrices, ISSN 2300-7451, Vol. 6, nr 1, s. 93-109Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In the spirit of our earlier paper [10] and Zhang andWang [16], we introduce the matrix of multiplicative q-Appell polynomials of order M is an element of Z. This is the representation of the respective q-Appell polynomials in ke-ke basis. Based on the fact that the q-Appell polynomials form a commutative ring [11], we prove that this set constitutes a q-Lie group with two dual q-multiplications in the sense of [9]. A comparison with earlier results on q-Pascal matrices gives factorizations according to [7], which are specialized to q-Bernoulli and q-Euler polynomials. We also show that the corresponding q-Bernoulli and q-Euler matrices form q-Lie subgroups. In the limit q -> 1 we obtain corresponding formulas for Appell polynomial matrices. We conclude by presenting the commutative ring of generalized q-Pascal functional matrices, which operates on all functions f is an element of C-q(infinity).

  • 13.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Matematik I-5.
    q-analogues of some operational formulas2004Rapport (Övrigt vetenskapligt)
  • 14.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    q-Bernoulli and q-Stirling numbers, an umbral approach2005Rapport (Övrigt vetenskapligt)
  • 15.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    q-Calculus as operational algebra2009Ingår i: Proceedings of the Estonian Academy of Sciences, ISSN 1736-6046, Vol. 58, nr 2, s. 73-97Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This second paper on operational calculus is a continuation of Ernst, T. q-Analogues of some operational formulas. Algebras Groups Geom., 2006, 23(4), 354-374. We find multiple q-analogues of formulas in Carlitz, L. A note on the Laguerre polynomials. Michigan Math. J., 1960, 7, 219-223, for the Cigler q-Laguerre polynomials (Ernst, T. A method for q-calculus. J. Nonlinear Math. Phys., 2003, 10(4), 487-525). The q-Jacobi polynomials (Jacobi, C. G. J. Werke 6. Berlin, 1891) are treated in the same way, we generalize further to q-analogues of Manocha, H. L. and Sharma, B. L. (Some formulae for Jacobi polynomials. Proc. Cambridge Philos. Soc., 1966, 62, 459-462) and Singh, R. P. (Operational formulae for Jacobi and other polynomials. Rend. Sem. Mat. Univ. Padova, 1965, 35, 237-244). A field of fractions for Cigler's multiplication operator (Cigler, J. Operatormethoden fur q-Identitaten II, q-Laguerre-Polynome. Monatsh. Math., 1981, 91, 105-117) is used in the computations. The formulas for q-Jacobi polynomials are mostly formal. We find q-orthogonality relations for q-Laguerre, q-Jacobi, and q-Legendre polynomials using q-integration by parts. This q-Legendre polynomial is given here for the first time, we also find its q-difference equations. An inequality for a q-exponential function is proved. The q-difference equation for (p)phi(p-1) (a(1),...,a(p); b(1),...,b(p-1)vertical bar q, z) is given improving on Smith, E. R. Zur Theorie der Heineschen Reihe und ihrer Verallgemeinerung. Diss. Univ. Munchen 1911, p. 11, by using e(k) = elementary symmetric polynomial. Partial q-difference equations for the q-Appell and q-Lauricella functions are found, improving on Jackson, F. H. On basic double hypergeometric functions. Quart. J. Math., Oxford Ser., 1942, 13, 69-82, and Gasper, G. and Rahman, M. Basic hypergeometric series. Second edition. Cambridge, 2004, p. 299, where q-difference equations for q-Appell functions were given with different notation. The q-difference equation for Phi(1) can also be written in canonical form, a q-analogue of [p. 146] Mellin, H. J. Uber den Zusammenhang zwischen den linearen Differential- und Differenzengleichunge, Acta Math., 1901, 25, 139-164.

  • 16.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    q-generating functions for one and two variables2002Rapport (Övrigt vetenskapligt)
  • 17.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Some new formulas involving Gamma q Functions2007Ingår i: Rendiconti del Seminario Matematico della Universita di Padova, ISSN 0041-8994, E-ISSN 2240-2926, Vol. 118, s. 159-188Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In a recent paper we found some new results for q-functions of many variables with the aid of the T, function. The Heine notation reminding of the hypergeometric case was used throughout, and some relations between Gamma(q) functions were presented. This paper aims at giving the promised longer exposition of Gamma(q)- revealing also the connection between this and the Jacobi-theta functions which appear in context. We will give a slightly generalized definition of the Heine series with more general tilde operators. 4 q-summation formulas of Andrews will be given in the new notation. The close affinity to q-binomial coefficient formulas will be stressed by expressing the finite q-hypergeometric formulas, the canonical form, in two ways. Two further q-analogues of Kummer's 2F(1)(-1) formula will be given. An ancient q-analogue of the Euler reflection formula will be used for the proof of a special case of the Bailey-Daum summation formula, conjectured in the previous paper.

  • 18.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Some new formulas involving Γq functions2006Rapport (Övrig (populärvetenskap, debatt, mm))
  • 19.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Some results for q-functions of many variables2004Rapport (Övrig (populärvetenskap, debatt, mm))
  • 20.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Some results for q-Laguerre polynomials2002Rapport (Övrigt vetenskapligt)
  • 21.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    The different tongues of q-calculus2008Ingår i: Proceedings of the Estonian Academy of Sciences, ISSN 1736-6046, Vol. 57, nr 2, s. 81-99Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this review paper we summarize the various dialects of q-calculus: quantum Calculus, time scales, and partitions. The close connection between Gamma q(x) functions on the one hand, and elliptic functions and theta functions on the other hand will be shown. The advantages of the Heine notation will be illustrated by the (q-)Euler reflection formula, q-Appell functions, Carlitz-AlSalam polynomials, and the so-called q-addition. We conclude with some short biographies about famous scientists in q-calculus.

  • 22.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    The history of q-calculus and a new method2000Licentiatavhandling, monografi (Övrigt vetenskapligt)
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