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Some results for qfunctions of many variables II
, 2004
"... This is a continuation of [19], where we presented an extension of the qhypergeometric function with connection to the title of this paper. In chapter one we present some quadratic q − hypergeometric transformations, to give more examples of this extension. In chapter two, systems of partial qdiff ..."
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This is a continuation of [19], where we presented an extension of the qhypergeometric function with connection to the title of this paper. In chapter one we present some quadratic q − hypergeometric transformations, to give more examples of this extension. In chapter two, systems of partial qdifference equations for the qAppell and qLauricella functions are presented in the authors notation. Other attempts to find these equations were made by Jackson. It turns out that these qdifference equations can be written in many equivalent forms, which gives rise to the notion of equivalence class for qdifference equations. In chapter three qanalogues of expansion formulas by Chaundy [11] and Burchnall & Chaundy [9] are found. In the process we obtain a corrected version of [26, (55) p.79]. In chapter four we find an expansion formula for a Φ 2:1 2:0 function by using Jackson’s sum of a terminating verywellpoised balanced 8φ7 series. We also find the corresponding qbinomial identity.
qBernoulli and qEuler Polynomials, an Umbral approach II
, 2009
"... We proceed with pseudoqAppell polynomials in the spirit of [12]. It turns out that these qBernoulli numbers are the same as BJHC,ν,q. As in [12] we find qanalogues of many formulas in [38], the umbral calculus works remarkably well also for pseudoqAppell pol., only the q is put up instead of d ..."
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We proceed with pseudoqAppell polynomials in the spirit of [12]. It turns out that these qBernoulli numbers are the same as BJHC,ν,q. As in [12] we find qanalogues of many formulas in [38], the umbral calculus works remarkably well also for pseudoqAppell pol., only the q is put up instead of down corresponding to inversion of basis. We also find new qEulerMaclaurin expansions.
Examples of qorthogonal polynomials
, 2008
"... We first study four different types of qHermite polynomials hν,q(x), ψν,q(x), kν,q(x), fν,q(x) from the point of view of generating functions and operational formulas. Three of these qHermite polynomials are qAppell polynomials, and the remaining one is a pseudo qAppell polynomial. qdifference ..."
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We first study four different types of qHermite polynomials hν,q(x), ψν,q(x), kν,q(x), fν,q(x) from the point of view of generating functions and operational formulas. Three of these qHermite polynomials are qAppell polynomials, and the remaining one is a pseudo qAppell polynomial. qdifference equations and qhypergeometric expressions are found. In this paper, we only obtain Rodriguez formulas for two of the four polynomials, namely hν,q(x) and kν,q(x), and accordingly we can only find qanalogues of the socalled Nielsen’s formula for these two polynomials. Matrix forms for the polynomials expressed by qPascal matrices are considered. An orthogonality relation for kν,q(x) expressed as a qintegral is found. The two Cigler qLaguerre polynomials are introduced and with the help of the relationship between hν,q(x) and the qLaguerre polynomials, new generating functions for hν,q(x) are found. In the second part of the paper we study orthogonality relations for qJacobi, qLaguerre and qLegendre polynomials. The proofs will all use qintegration by parts, a method equivalent to the previously used recurrence technique. The orthogonality relations are all of discrete type.
qDeformed Matrix PseudoGroups
, 2009
"... The quantum groups and Hopf algebras have invaded theoretical physics. We give an alternate (and possibly more understandable) description of these phenomena in this paper. In another paper [9] the author has introduced a qumbral calculus in the spirit of Rota. This umbral calculus contains two du ..."
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The quantum groups and Hopf algebras have invaded theoretical physics. We give an alternate (and possibly more understandable) description of these phenomena in this paper. In another paper [9] the author has introduced a qumbral calculus in the spirit of Rota. This umbral calculus contains two dual q–additions, the Nalli–Ward–Alsalam (NWA) q–addition and the Jackson–Hahn–Cigler (JHC) q–addition. In a further paper [10], a matrix calculus was built up from the Jackson qexponential function. It is wellknown that Lie groups (oneparameter subgroups) can be made up from the exponential function applied to the Lie algebra (e.g. Pauli matrices). A combination of these two techniques will lead to a new and natural qdeformation of matrix Lie groups. We will use two matrix multiplications, the first one is ordinary multiplication, and the second one contains an involution operator τ: q → 1 q. We supply these two matrix multiplications with an attractive associative structure. It turns out that these deformed matrix pseudo–groups have quite a different character than the wellknown quantum groups. In spite of this, we have decided to keep the same notation for these qdeformed matrix groups as for the corresponding quantum groups. The first matrix pseudogroup to be defined SOq(2), is not seen so often explicitly in the quantum group context. However several physicists use the corresponding SOq(3) symmetric quantum mechanics [11], [30].
Available online at www.eap.ee/proceedings The different tongues of qcalculus
, 2008
"... Abstract. In this review paper we summarize the various dialects of qcalculus: quantum calculus, time scales, and partitions. The close connection between Γq(x) functions on the one hand, and elliptic functions and theta functions on the other hand will be shown. The advantages of the Heine notatio ..."
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Abstract. In this review paper we summarize the various dialects of qcalculus: quantum calculus, time scales, and partitions. The close connection between Γq(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, qAppell functions, Carlitz– AlSalam polynomials, and the socalled qaddition. We conclude with some short biographies about famous scientists in qcalculus. Key words: elliptic functions, theta functions, qAppell functions, qaddition, Carlitz–AlSalam polynomial. 1.