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The Askeyscheme of hypergeometric orthogonal polynomials and its qanalogue
, 1998
"... We list the socalled Askeyscheme of hypergeometric orthogonal polynomials and we give a q analogue of this scheme containing basic hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation, the second order di#erent ..."
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Cited by 376 (4 self)
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We list the socalled Askeyscheme of hypergeometric orthogonal polynomials and we give a q analogue of this scheme containing basic hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation, the second order di#erential or di#erence equation, the forward and backward shift operator, the Rodriguestype formula and generating functions of all classes of orthogonal polynomials in this scheme. In chapter 2 we give the limit relations between di#erent classes of orthogonal polynomials listed in the Askeyscheme. In chapter 3 we list the qanalogues of the polynomials in the Askeyscheme. We give their definition, orthogonality relation, three term recurrence relation, second order di#erence equation, forward and backward shift operator, Rodriguestype formula and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally, in chapter 5 we...
Adaptive Modulation over Nakagami Fading Channels
, 1998
"... We first study the capacity of Nakagami multipath fading (NMF) channels with an average power constraint for three power and rate adaptation policies. We obtain closedform solutions for NMF channel capacity for each power and rate adaptation strategy. Results show that rate adaptation is the key to ..."
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Cited by 75 (5 self)
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We first study the capacity of Nakagami multipath fading (NMF) channels with an average power constraint for three power and rate adaptation policies. We obtain closedform solutions for NMF channel capacity for each power and rate adaptation strategy. Results show that rate adaptation is the key to increasing link spectral efficiency. We analyze therefore the performance of constantpower variablerate MQAM schemes over NMF channels. We obtain closedform expressions for the outage probability, spectral efficiency and average biterrorrate (BER) assuming perfect channel estimation and negligible time delay. We also analyze the impact of time delay on the BER of adaptive MQAM. Keywords Link Spectral Efficiency, Adaptive Modulation Techniques, and Nakagami Fading. I. Introduction The radio spectrum available for wireless services is extremely scarce, while demand for these services is growing at a rapid pace [1]. Hence spectral efficiency is of primary concern in the design of fut...
Orthogonal polynomials for exponential weights x2ρe−2Q(x) on [0,d
 J. Approx. Theory
"... xi + 476 pp With each of a large class of positive measures µ on the real line it is possible to associate a sequence {pn(x)} of orthogonal polynomials with the property that pm(x)pn(x)dµ(x) = 0, m ̸ = n, 1, m = n. (1) Such a sequence satisfies a three term recurrence relation xpn(x) = anPn+1(x) + ..."
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Cited by 47 (15 self)
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xi + 476 pp With each of a large class of positive measures µ on the real line it is possible to associate a sequence {pn(x)} of orthogonal polynomials with the property that pm(x)pn(x)dµ(x) = 0, m ̸ = n, 1, m = n. (1) Such a sequence satisfies a three term recurrence relation xpn(x) = anPn+1(x) + bnPn(x) + an−1Pn−1(x) (2) Conversely, for suitable starting values and coefficient sequences {an}, {bn}, the recurrence relation (2) generates a sequence of polynomials satisfying (1) for some measure µ. The polynomials have their zeros within the interval of support of the measure. Examples date from the 19th century. The Jacobi polynomials P (α,β) n (x) are orthogonal with respect to µ with support [−1, 1], where dµ = (1 − x) α+1 (1 +
Algebraic transformations of Gauss hypergeometric functions, preprint
, 2004
"... The paper classifies algebraic transformations of Gauss hypergeometric functions and pullback transformations between hypergeometric differential equations. This classification recovers the classical transformations of degree 2, 3, 4, 6, and finds other transformations of some special classes of th ..."
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Cited by 23 (9 self)
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The paper classifies algebraic transformations of Gauss hypergeometric functions and pullback transformations between hypergeometric differential equations. This classification recovers the classical transformations of degree 2, 3, 4, 6, and finds other transformations of some special classes of the Gauss hypergeometric function.
Impact of channel prediction on adaptive coded modulation performance in Rayleigh fading
 IEEE Transactions on Vehicular Technology
, 2004
"... Adaptive coded modulation (ACM) is a promising tool for increasing the spectral efficiency on timevarying mobile channels while keeping a predictable bit error rate. An important restriction in systems with such a transmission scheme is that the transmitter needs to have accurate channel state info ..."
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Cited by 17 (3 self)
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Adaptive coded modulation (ACM) is a promising tool for increasing the spectral efficiency on timevarying mobile channels while keeping a predictable bit error rate. An important restriction in systems with such a transmission scheme is that the transmitter needs to have accurate channel state information (CSI). Earlier analysis of ACM systems usually assume that the transmitter has perfect knowledge of the channel, or that the CSI is accurate but outdated. In this paper, we investigate the effects of predicting the CSI using a linear fading envelope predictor, in order to enhance the performance of an ACM system. For the case when multidimensional trellis codes are used on Rayleigh fading channels, we obtain approximative closedform expressions for bit error rate and average spectral efficiency. Numerical examples are given for the case of Jakes correlation profile and maximum a posteriorioptimal predictor coefficients.
The Incomplete Gamma Functions Since Tricomi
 In Tricomi's Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147, Accademia Nazionale dei Lincei
, 1998
"... The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asy ..."
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Cited by 15 (1 self)
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The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.
Adaptive MQam Modulation over Nakagami Fading Channels
 in IEEE Global Communications Conference
, 1997
"... We study the performance of constantpower variablerate MQAM schemes over Nakagami fading channels. We obtain closedform expressions for the outage probability, spectral efficiency and average biterrorrate (BER) assuming perfect channel estimation and negligible time delay. For a target BER of ..."
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Cited by 12 (2 self)
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We study the performance of constantpower variablerate MQAM schemes over Nakagami fading channels. We obtain closedform expressions for the outage probability, spectral efficiency and average biterrorrate (BER) assuming perfect channel estimation and negligible time delay. For a target BER of 10 \Gamma3 , the spectral efficiency of adaptive continuous rate MQAM comes within 5 dB of the Shannon capacity limit, and adaptive discrete rate MQAM comes within 6.2 dB of this limit. Nonadaptive BPSK suffers a large spectral efficiency penalty relative to these adaptive techniques. We also analyze the impact of time delay on the BER of adaptive MQAM. Results show that systems with low BER requirements will be more sensitive to time delay, but will still operate satisfactorily if the delay is below a critical value. 1. INTRODUCTION The radio spectrum available for wireless services is extremely scarce, while demand for these services is growing at a rapid pace [1]. Spectral efficien...
8 Lectures on Quantum Groups and qSpecial Functions
, 1996
"... The lecture notes contains an introduction to quantum groups, qspecial functions and their interplay. After generalities on Hopf algebras, orthogonal polynomials and basic hypergeometric series we work out the relation between the quantum SU(2) group and the AskeyWilson polynomials out in detail ..."
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Cited by 9 (2 self)
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The lecture notes contains an introduction to quantum groups, qspecial functions and their interplay. After generalities on Hopf algebras, orthogonal polynomials and basic hypergeometric series we work out the relation between the quantum SU(2) group and the AskeyWilson polynomials out in detail as the main example. As an application we derive an addition formula for a twoparameter subfamily of AskeyWilson polynomials. A relation between the AlSalam and Chihara polynomials and the quantised universal enveloping algebra for su(1, 1) is given. Finally, more examples and other approaches as well as some open problems are given.
On Polynomials Related with HermitePadé Approximations to the Exponential Function
"... We investigate the polynomials P n ; Qm and R s , having degrees n; m and s respectively, with P n monic, that solve the approximation problem Enms (x) := P n (x)e \Gamma2x +Qm (x)e \Gammax +R s (x) = O(x n+m+s+2 ) as x ! 0: We give a connection between the coefficients of each of the polynomi ..."
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Cited by 7 (2 self)
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We investigate the polynomials P n ; Qm and R s , having degrees n; m and s respectively, with P n monic, that solve the approximation problem Enms (x) := P n (x)e \Gamma2x +Qm (x)e \Gammax +R s (x) = O(x n+m+s+2 ) as x ! 0: We give a connection between the coefficients of each of the polynomials P n ; Qm and R s and certain hypergeometric functions, which leads to a simple expression for Qm in the case n = s. The approximate location of the zeros of Qm , when n AE m and n = s, are deduced from the zeros of the classical Hermite polynomial. Contour integral representations of P n ; Qm ; R s and Enms are given and, using saddle point methods, we derive the exact asymptotics of P n ; Qm and R s as n; m and s tend to infinity through certain ray sequences. We also discuss aspects of the more complicated uniform asymptotic methods for obtaining insight into the zero distribution of the polynomials, and we give an example showing the zeros of the polynomials P n ; Qm and R s for the case n = s = 40; m = 45. 1991 Mathematics Subject Classification: 41A21, 30E15, 30C15, 41A60. Keywords & Phrases: HermitePad'e approximant, exponential function, hypergeometric series, error asymptotics, zero distribution, HermitePad'e Type I polynomials. Note: Work carried out under project MAS2.8 Exploratory research. 1.
2000), Numerical and asymptotic aspects of parabolic cylinder functions
 J. Comp. Appl. Math
"... Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic expansions given earlier by F.W.J. Olver. Some of his results are mo ..."
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Cited by 7 (6 self)
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Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic expansions given earlier by F.W.J. Olver. Some of his results are modified to improve the asymptotic properties and to enlarge the intervals for using the expansions in numerical algorithms. Olver’s results are obtained from the differential equation of the parabolic cylinder functions; we mention how modified expansions can be obtained from integral representations. Numerical tests are given for three expansions in terms of elementary functions. In this paper only real values of the parameters will be considered. 1991 Mathematics Subject Classification: 33C15, 41A60, 65D20.