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Journal articles on the topic 'Jacobi polynomials'

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Published: 4 June 2021
Last updated: 25 January 2023

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1

Lukš, Antonín. "The quantized Jacobi polynomials." Applications of Mathematics 32, no. 6 (1987): 417–26. http://dx.doi.org/10.21136/am.1987.104273.

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2

Stojanovic, Nikola, and Negovan Stamenkovic. "Lowpass filters approximation based on the Jacobi polynomials." Facta universitatis - series: Electronics and Energetics 30, no. 3 (2017): 351–62. http://dx.doi.org/10.2298/fuee1703351s.

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A case study related to the design the analog lowpass filter using a set of orthogonal Jacobi polynomials, having four parameters to vary, is considered. The Jacobi polynomial has been modified in order to be used as a filter approximating function. The obtained magnitude response is more general than the response of the classical ultra-spherical filter, due to one additional parameter available in orthogonal Jacobi polynomials. This additional parameter may be used to obtain a magnitude response having either smaller passband ripple, smaller group delay variation or sharper cutoff slope. Two methods for transfer function approximation are investigated: the first method is based on the known shifted Jacobi polynomial, and the second method is based on the proposed modification of Jacobi polynomials. The shifted Jacobi polynomials are suitable only for odd degree transfer function. However, the proposed modified Jacobi polynomial filter function is more general but not orthogonal. It is transformed into orthogonal polynomial when orders are equal and then includes the Chebyshev filter of the first kind, the Chebyshev filter of the second kind, the Legendre filter, Gegenbauer (ultraspherical) filter and many other filters, as its special cases.
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3

Bhrawy, A. H., and A. S. Alofi. "An Accurate Spectral Galerkin Method for Solving Multiterm Fractional Differential Equations." Mathematical Problems in Engineering 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/728736.

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This paper reports a new formula expressing the Caputo fractional derivatives for any order of shifted generalized Jacobi polynomials of any degree in terms of shifted generalized Jacobi polynomials themselves. A direct solution technique is presented for solving multiterm fractional differential equations (FDEs) subject to nonhomogeneous initial conditions using spectral shifted generalized Jacobi Galerkin method. The homogeneous initial conditions are satisfied exactly by using a class of shifted generalized Jacobi polynomials as a polynomial basis of the truncated expansion for the approximate solution. The approximation of the spatial Caputo fractional order derivatives is expanded in terms of a class of shifted generalized Jacobi polynomialsJnα,−β(x)withx∈(0,1), andnis the polynomial degree. Several numerical examples with comparisons with the exact solutions are given to confirm the reliability of the proposed method for multiterm FDEs.
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4

He, M. X., and P. Natalini. "Relativistic jacobi polynomials." Integral Transforms and Special Functions 8, no. 1-2 (July 1999): 43–56. http://dx.doi.org/10.1080/10652469908819215.

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5

Bonneux, Niels. "Exceptional Jacobi polynomials." Journal of Approximation Theory 239 (March 2019): 72–112. http://dx.doi.org/10.1016/j.jat.2018.11.002.

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6

Amir, Saba. "Extended Jacobi polynomials." International Journal of Contemporary Mathematical Sciences 9 (2014): 535–44. http://dx.doi.org/10.12988/ijcms.2014.4778.

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7

Rababah, Abedallah. "Jacobi-Bernstein Basis Transformation." Computational Methods in Applied Mathematics 4, no. 2 (2004): 206–14. http://dx.doi.org/10.2478/cmam-2004-0012.

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Abstract In this paper we derive the matrix of transformation of the Jacobi polynomial basis form into the Bernstein polynomial basis of the same degree n and vice versa. This enables us to combine the superior least-squares performance of the Jacobi polynomials with the geometrical insight of the Bernstein form. Application to the inversion of the Bézier curves is given.
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8

Gustavsson, Jan. "Some sums of Legendre and Jacobi polynomials." Mathematica Bohemica 126, no. 1 (2001): 141–49. http://dx.doi.org/10.21136/mb.2001.133910.

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9

Zayed, Ahmed I. "Jacobi polynomials as generalized Faber polynomials." Transactions of the American Mathematical Society 321, no. 1 (January 1, 1990): 363–78. http://dx.doi.org/10.1090/s0002-9947-1990-0965745-1.

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10

Ismail, Mourad E. H. "Relativistic orthogonal polynomials are Jacobi polynomials." Journal of Physics A: Mathematical and General 29, no. 12 (June 21, 1996): 3199–202. http://dx.doi.org/10.1088/0305-4470/29/12/023.

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11

Boubaker, K. "A Confirmed Model to Polymer Core-Shell Structured Nanofibers Deposited via Coaxial Electrospinning." ISRN Polymer Science 2012 (October 14, 2012): 1–6. http://dx.doi.org/10.5402/2012/603108.

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A model to core-shell structured polymer nanofibers deposited via coaxial electrospinning is presented. Investigations are based on a modified Jacobi-Gauss collocation spectral method, proposed along with the Boubaker Polynomials Expansion Scheme (BPES), for providing solution to a nonlinear Lane-Emden-type equation. The spatial approximation has been based on shifted Jacobi polynomials with was n the polynomial degree. The Boubaker Polynomials Expansion Scheme (BPES) main features, concerning the embedded boundary conditions, have been outlined. The modified Jacobi-Gauss points are used as collocation nodes. Numerical examples are included to demonstrate the validity and applicability of the technique, and a comparison is made with existing results. It has been revealed that both methods are easy to implement and yield very accurate results.
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12

SHARMA, S. K., and U. KHANAL. "PERTURBATION OF FRW SPACETIME IN NP FORMALISM." International Journal of Modern Physics D 23, no. 02 (January 29, 2014): 1450017. http://dx.doi.org/10.1142/s0218271814500175.

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Jacobi polynomials appear to play a very important role in describing all the spin field (s = 0, 1/2, 1, 2) perturbation of the Friedmann–Robertson–Walker (FRW) spacetime. The formulation becomes very transparent when done in Newman–Penrose (NP) formalism. All the variables are separable, and the spatial eigenfunctions turn out to be Jacobian polynomials in different forms. In particular, the angular ones are expressible as spin weighted spherical harmonics which are just the spherical harmonics formed with Jacobi polynomials. The radial eigenfunctions are also Jacobi polynomials but with unconventional parameters. Various properties of these polynomials are used to describe the scalar, vector and tensor modes of the perturbation. The Green's function of the scalar perturbations and also its Lienard–Wiechert type potentials are derived, and are shown to reduce to the familiar ones in the limit to flat FRW case. Some of the components of the perturbed metric tensor hμν have also been calculated.
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13

Koelink, H. T. "Addition Formula For Big q-Legendre Polynomials From The Quantum Su(2) Group." Canadian Journal of Mathematics 47, no. 2 (April 1, 1995): 436–48. http://dx.doi.org/10.4153/cjm-1995-024-8.

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AbstractFrom Koornwinder's interpretation of big q-Legendre polynomials as spherical elements on the quantum SU(2) group an addition formula is derived for the big g-Legendre polynomial. The formula involves Al-Salam-Carlitz polynomials, little q-Jacobi polynomials and dual q-Krawtchouk polynomials. For the little q-ultraspherical polynomials a product formula in terms of a big q-Legendre polynomial follows by q-integration. The addition and product formula for the Legendre polynomials are obtained when q tends to 1.
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14

Imani, Ahmad, Azim Aminataei, and Ali Imani. "Collocation Method via Jacobi Polynomials for Solving Nonlinear Ordinary Differential Equations." International Journal of Mathematics and Mathematical Sciences 2011 (2011): 1–11. http://dx.doi.org/10.1155/2011/673085.

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We extend a collocation method for solving a nonlinear ordinary differential equation (ODE) via Jacobi polynomials. To date, researchers usually use Chebyshev or Legendre collocation method for solving problems in chemistry, physics, and so forth, see the works of (Doha and Bhrawy 2006, Guo 2000, and Guo et al. 2002). Choosing the optimal polynomial for solving every ODEs problem depends on many factors, for example, smoothing continuously and other properties of the solutions. In this paper, we show intuitionally that in some problems choosing other members of Jacobi polynomials gives better result compared to Chebyshev or Legendre polynomials.
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15

Bedoya, D., M. Ortega, W. Ramírez, and A. Urieles. "New Biparametric Families of Apostol-Frobenius-Euler Polynomials level-m." Matematychni Studii 55, no. 1 (March 4, 2021): 10–23. http://dx.doi.org/10.30970/ms.55.1.10-23.

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We introduce two biparametric families of Apostol-Frobenius-Euler polynomials of level-$m$. We give some algebraic properties, as well as some other identities which connect these polynomial class with the generalized $\lambda$-Stirling type numbers of the second kind, the generalized Apostol--Bernoulli polynomials, the generalized Apostol--Genocchi polynomials, the generalized Apostol--Euler polynomials and Jacobi polynomials. Finally, we will show the differential properties of this new family of polynomials.
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16

Durán, Antonio J., and Manuel D. de la Iglesia. "Bispectral Jacobi type polynomials." Advances in Applied Mathematics 136 (May 2022): 102322. http://dx.doi.org/10.1016/j.aam.2022.102322.

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17

Bouras, B. "Generalized Jacobi orthogonal polynomials." Integral Transforms and Special Functions 18, no. 10 (October 2007): 715–30. http://dx.doi.org/10.1080/10652460701445666.

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18

Grünbaum, F. Alberto. "Matrix valued Jacobi polynomials." Bulletin des Sciences Mathématiques 127, no. 3 (May 2003): 207–14. http://dx.doi.org/10.1016/s0007-4497(03)00009-5.

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19

Atia, M. J., J. Alaya, and A. Ronveaux. "Some generalized Jacobi polynomials." Computers & Mathematics with Applications 45, no. 4-5 (February 2003): 843–50. http://dx.doi.org/10.1016/s0898-1221(03)00045-2.

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20

Gautschi, Walter. "Sub-range Jacobi polynomials." Numerical Algorithms 61, no. 4 (March 14, 2012): 649–57. http://dx.doi.org/10.1007/s11075-012-9556-z.

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21

Vinet, Luc, Guo-Fu Yu, and Alexei Zhedanov. "$-1$ Krall–Jacobi polynomials." Methods and Applications of Analysis 22, no. 3 (2015): 249–58. http://dx.doi.org/10.4310/maa.2015.v22.n3.a1.

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22

Hendriksen, Erik. "Associated Jacobi-Laurent polynomials." Journal of Computational and Applied Mathematics 32, no. 1-2 (November 1990): 125–41. http://dx.doi.org/10.1016/0377-0427(90)90424-x.

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23

Haagerup, Uffe, and Henrik Schlichtkrull. "Inequalities for Jacobi polynomials." Ramanujan Journal 33, no. 2 (July 2, 2013): 227–46. http://dx.doi.org/10.1007/s11139-013-9472-4.

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24

Atia, Mohamed Jalel. "An example of nonsymmetric semi-classical form of classs=1; generalization of a case of Jacobi sequence." International Journal of Mathematics and Mathematical Sciences 24, no. 10 (2000): 673–89. http://dx.doi.org/10.1155/s0161171200004671.

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We give explicitly the recurrence coefficients of a nonsymmetric semi-classical sequence of polynomials of classs=1. This sequence generalizes the Jacobi polynomial sequence, that is, we give a new orthogonal sequence{Pˆn(α,α+1)(x,μ)}, whereμis an arbitrary parameter withℜ(1−μ)>0in such a way that forμ=0one has the well-known Jacobi polynomial sequence{Pˆn(α,α+1)(x)},n≥0.
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25

ISMAIL, MOURAD E. H., and ERIK KOELINK. "SPECTRAL PROPERTIES OF OPERATORS USING TRIDIAGONALIZATION." Analysis and Applications 10, no. 03 (July 2012): 327–43. http://dx.doi.org/10.1142/s0219530512500157.

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A general scheme for tridiagonalizing differential, difference or q-difference operators using orthogonal polynomials is described. From the tridiagonal form the spectral decomposition can be described in terms of the orthogonality measure of generally different orthogonal polynomials. Three examples are worked out: (1) related to Jacobi and Wilson polynomials for a second order differential operator, (2) related to little q-Jacobi polynomials and Askey–Wilson polynomials for a bounded second order q-difference operator, (3) related to little q-Jacobi polynomials for an unbounded second order q-difference operator. In case (1) a link with the Jacobi function transform is established, for which we give a q-analogue using example (2).
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26

Nevai, Paul, Tamás Erdélyi, and Alphonse P. Magnus. "Generalized Jacobi Weights, Christoffel Functions, and Jacobi Polynomials." SIAM Journal on Mathematical Analysis 25, no. 2 (March 1994): 602–14. http://dx.doi.org/10.1137/s0036141092236863.

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27

XU, XUE-XIANG, LI-YUN HU, and HONG-YI FAN. "ON THE NORMALIZED TWO-MODE PHOTON-SUBTRACTED SQUEEZED VACUUM STATE." Modern Physics Letters A 24, no. 32 (October 20, 2009): 2623–30. http://dx.doi.org/10.1142/s0217732309031168.

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We show that the two-mode photon-subtracted squeezed state (TPSSS) is a squeezed two-variable Hermite polynomial excitation state, and we can therefore determine its normalization as a Jacobi polynomial of the squeezing parameter. Some new relations about the Jacobi polynomials are obtained by this analysis. We also show that the TPSSS can be treated as a two-variable Hermite-polynomial excitation on squeezed vacuum state. The technique of integration within an ordered product of operators brings convenience in our derivation.
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28

Rahman, Mizan. "A q-Extension of Feldheim's Bilinear Sum for Jacobi Polynomials and Some Applications." Canadian Journal of Mathematics 37, no. 3 (June 1, 1985): 551–76. http://dx.doi.org/10.4153/cjm-1985-030-0.

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The main objective of this paper is to find useful q-extensions of Feldheim's [6] bilinear formula for Jacobi polynomials, namely,1.1where the Appel function F4 is defined by1.2α1, α2, ρ are arbitrary complex parameters such that the series on both sides of (1.1) are convergent, and1.3is the Jacobi polynomial of degree k, (a)k being the usual shifted factorial.
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29

Abd-Elhameed, Waleed Mohamed, and Badah Mohamed Badah. "New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas." Mathematics 9, no. 13 (July 4, 2021): 1573. http://dx.doi.org/10.3390/math9131573.

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This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi polynomials are expressed in terms of other shifted Jacobi polynomials. The derived moments formula involves a hypergeometric function of the type 4F3(1), which cannot be summed in general, but for special choices of the involved parameters, it can be summed. The reduced moments formulas lead to establishing new linearization formulas of certain parameters of Jacobi polynomials. Another approach for obtaining other linearization formulas of some Jacobi polynomials depends on making use of the connection formulas between two different Jacobi polynomials. In the two suggested approaches, we utilize some standard reduction formulas for certain hypergeometric functions of the unit argument such as Watson’s and Chu-Vandermonde identities. Furthermore, some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij may be utilized for the same purpose. As an application of some of the derived linearization formulas, we propose a numerical algorithm to solve the non-linear Riccati differential equation based on the application of the spectral tau method.
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30

Jafarov, Elchin I., Aygun M. Mammadova, and Joris Van der Jeugt. "On the Direct Limit from Pseudo Jacobi Polynomials to Hermite Polynomials." Mathematics 9, no. 1 (January 4, 2021): 88. http://dx.doi.org/10.3390/math9010088.

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In this short communication, we present a new limit relation that reduces pseudo-Jacobi polynomials directly to Hermite polynomials. The proof of this limit relation is based upon 2F1-type hypergeometric transformation formulas, which are applicable to even and odd polynomials separately. This limit opens the way to studying new exactly solvable harmonic oscillator models in quantum mechanics in terms of pseudo-Jacobi polynomials.
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31

Stokman, Jasper V., and Tom H. Koornwinder. "Limit Transitions for BC Type Multivariable Orthogonal Polynomials." Canadian Journal of Mathematics 49, no. 2 (April 1, 1997): 374–405. http://dx.doi.org/10.4153/cjm-1997-019-9.

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AbstractLimit transitions will be derived between the five parameter family of Askey-Wilson polynomials, the four parameter family of big q-Jacobi polynomials and the three parameter family of little q-Jacobi polynomials in n variables associated with root system BC. These limit transitions generalize the known hierarchy structure between these families in the one variable case. Furthermore it will be proved that these three families are q-analogues of the three parameter family of BC type Jacobi polynomials in n variables. The limit transitions will be derived by taking limits of q-difference operators which have these polynomials as eigenfunctions.
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32

Milovanovic, Gradimir, and Aleksandar Cvetkovic. "Complex Jacobi matrices and quadrature rules." Filomat, no. 17 (2003): 117–34. http://dx.doi.org/10.2298/fil0317117m.

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Given any sequence of orthogonal polynomials, satisfying the three term recurrence relation xpn(x) = ?n+1pn+1(x) + ?npn(x) + ?npn-1(x), p-1(x)=0 po(x)=1 with ?n ? 0, n ? N, ?0 = 1, an infinite Jacobi matrix can be associated in the following way ? ? ??0 ?1 0...? ? ? ??1 ?1 ?2...? ? ? J = ?0 ?2 ?2...? ?.... ? ?.... ? ?....? ? ? In the general case if the sequences {?n} or {?n} are complex the associated Jacobi matrix is complex. Under the condition that both sequences {?n}and {?n} are uniformly bounded, the associated Jacobi matrix can be understood as a linear operator J acting on ?2, the space of all complex square-summable sequences, where the value of the operator J at the vector x is a product of an infinite vector x and an infinite matrix J in the matrix sense. The case when the sequences {?n} and {?n} are not uniformly bounded, an operator acting on ?2 can not be defined that easily. Additional properties of the sequence of orthogonal polynomials are needed in order to be able to define the operator uniquely. The case when the sequences ?n and ?n are real is very well understood. The spectra of the Jacobi matrix J equals the support of the measure of orthogonality for the given sequence of orthogonal polynomials. All zeros of orthogonal polynomials are real, simple and interlace, contained in the convex hull of the spectra of the Jacobi operator associated with the infinite Jacobi matrix J. Every point in ?(J) attracts zeros of orthogonal polynomials. An application of orthogonal polynomials is the construction of quadrature rules for the approximation of integration with respect to the measure of orthogonality. For arbitrary sequences {?n} and {?n} the situation is changed dramatically. Zeros of orthogonal polynomials need not be simple; they are not real and they do not necessarily lie in the convex hull of ?(J). There is also a little known about convergence results of related quadrature rule. Only in recent years a connection between complex Jacobi matrices and related orthogonal polynomials is interesting again (see [2]). Studies of complex Jacobi operators should lead to a better understanding of related orthogonal polynomials, but also the study of orthogonal polynomials with the complex Jacobi matrices should put more light on the non-hermitian banded symmetric matrices. In this lecture some results are given about complex Jacobi matrices and related quadrature rules, and also some interesting examples are presented.
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33

De Sole, Alberto, and Victor G. Kac. "Subalgebras of gcN and Jacobi Polynomials." Canadian Mathematical Bulletin 45, no. 4 (December 1, 2002): 567–605. http://dx.doi.org/10.4153/cmb-2002-055-9.

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AbstractWe classify the subalgebras of the general Lie conformal algebra gcN that act irreducibly on [∂]N and that are normalized by the sl2-part of a Virasoro element. The problem turns out to be closely related to classical Jacobi polynomials , σ ∈ . The connection goes both ways—we use in our classification some classical properties of Jacobi polynomials, and we derive from the theory of conformal algebras some apparently new properties of Jacobi polynomials.
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34

BERCEANU, STEFAN. "A HOLOMORPHIC REPRESENTATION OF THE JACOBI ALGEBRA." Reviews in Mathematical Physics 18, no. 02 (March 2006): 163–99. http://dx.doi.org/10.1142/s0129055x06002619.

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A representation of the Jacobi algebra 𝔥1 ⋊ 𝔰𝔲(1, 1) by first-order differential operators with polynomial coefficients on the manifold [Formula: see text] is presented. The Hilbert space of holomorphic functions on which the holomorphic first-order differential operators with polynomials coefficients act is constructed.
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35

Varma, Serhan, and Fatma Taşdelen. "Biorthogonal matrix polynomials related to Jacobi matrix polynomials." Computers & Mathematics with Applications 62, no. 10 (November 2011): 3663–68. http://dx.doi.org/10.1016/j.camwa.2011.08.063.

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36

Rababah, A., and M. Alqudah. "Jacobi-weighted orthogonal polynomials on triangular domains." Journal of Applied Mathematics 2005, no. 3 (2005): 205–17. http://dx.doi.org/10.1155/jam.2005.205.

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We construct Jacobi-weighted orthogonal polynomials𝒫n,r(α,β,γ)(u,v,w),α,β,γ>−1,α+β+γ=0, on the triangular domainT. We show that these polynomials𝒫n,r(α,β,γ)(u,v,w)over the triangular domainTsatisfy the following properties:𝒫n,r(α,β,γ)(u,v,w)∈ℒn,n≥1,r=0,1,…,n,and𝒫n,r(α,β,γ)(u,v,w)⊥𝒫n,s(α,β,γ)(u,v,w)forr≠s. And hence,𝒫n,r(α,β,γ)(u,v,w),n=0,1,2,…,r=0,1,…,nform an orthogonal system over the triangular domainTwith respect to the Jacobi weight function. These Jacobi-weighted orthogonal polynomials on triangular domains are given in Bernstein basis form and thus preserve many properties of the Bernstein polynomial basis.
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37

Yadav, Sarjoo Prasad, Rakesh Kumar Yadav, and Dinesh Kumar Yadav. "On the Nörlund Method of Signal Processing Involving Coifman Wavelets." Advanced Materials Research 433-440 (January 2012): 3378–87. http://dx.doi.org/10.4028/www.scientific.net/amr.433-440.3378.

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We consider on real line R a space of signals which are p-power (1 ≤ p ≤∞ ) Lebesgue integrable with weight w(x) = (1 - x)α (1 + x)β , ( α, β > -1) on [-1, 1] R. A subspace χabNvof Xabvis recognized by restricting the types of signals, so that the signals are represented by Jacobi Polynomials. Then by the derivability of Jacobi polynomials, we reach to the conclusion that the signals of the subspace XαβNv can be represented by the Coifman wavelets. The method involves the N rlund summation of Fourier-Jacobi expansions and the properties of Jacobi polynomials in [--1, 1] R
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38

Nadarajah, Saralees. "Alternative Formula for Jacobi Polynomials." Missouri Journal of Mathematical Sciences 20, no. 2 (May 2008): 150–52. http://dx.doi.org/10.35834/mjms/1316032815.

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39

Shrivastava, H. S. P. "On Multiindex Multivariable Jacobi Polynomials." Integral Transforms and Special Functions 13, no. 5 (January 1, 2002): 417–45. http://dx.doi.org/10.1080/10652460213529.

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40

Genest, Vincent X., Jean-Michel Lemay, Luc Vinet, and Alexei Zhedanov. "Two-variable -1 Jacobi polynomials." Integral Transforms and Special Functions 26, no. 6 (February 19, 2015): 411–25. http://dx.doi.org/10.1080/10652469.2015.1013034.

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41

Chen, Yang, and Mourad Ismail. "Jacobi polynomials from compatibility conditions." Proceedings of the American Mathematical Society 133, no. 2 (August 30, 2004): 465–72. http://dx.doi.org/10.1090/s0002-9939-04-07566-5.

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42

Postelmans, Kelly, and Walter Van Assche. "Multiple little q-Jacobi polynomials." Journal of Computational and Applied Mathematics 178, no. 1-2 (June 2005): 361–75. http://dx.doi.org/10.1016/j.cam.2004.03.031.

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43

Sauer, Tomas. "Jacobi polynomials in Bernstein form." Journal of Computational and Applied Mathematics 199, no. 1 (February 2007): 149–58. http://dx.doi.org/10.1016/j.cam.2005.07.028.

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44

Chen, Li-Chen, and Mourad E. H. Ismail. "On Asymptotics of Jacobi Polynomials." SIAM Journal on Mathematical Analysis 22, no. 5 (September 1991): 1442–49. http://dx.doi.org/10.1137/0522092.

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45

Bouzeffour, Fethi, and Mubariz Garayev. "Multiple big q-Jacobi polynomials." Bulletin of Mathematical Sciences 10, no. 02 (May 19, 2020): 2050013. http://dx.doi.org/10.1142/s1664360720500137.

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Here, we investigate type II multiple big [Formula: see text]-Jacobi orthogonal polynomials. We provide their explicit formulae in terms of basic hypergeometric series, raising and lowering operators, Rodrigues formulae, third-order [Formula: see text]-difference equation, and we obtain recurrence relations.
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46

Letessier, Jean. "Co-recursive associated Jacobi polynomials." Journal of Computational and Applied Mathematics 57, no. 1-2 (February 1995): 203–13. http://dx.doi.org/10.1016/0377-0427(93)e0246-i.

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47

Znojil, Miloslav. "Jacobi polynomials and bound states." Journal of Mathematical Chemistry 19, no. 2 (1996): 205–13. http://dx.doi.org/10.1007/bf01165184.

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48

Nuttall, J. "Asymptotics of generalized jacobi polynomials." Constructive Approximation 2, no. 1 (December 1986): 59–77. http://dx.doi.org/10.1007/bf01893417.

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49

Pijeira-Cabrera, Hector, and Daniel Rivero-Castillo. "Iterated Integrals of Jacobi Polynomials." Bulletin of the Malaysian Mathematical Sciences Society 43, no. 3 (September 14, 2019): 2745–56. http://dx.doi.org/10.1007/s40840-019-00831-8.

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50

Abdelkawy, M. A. "A Collocation Method Based on Jacobi and Fractional Order Jacobi Basis Functions for Multi-Dimensional Distributed-Order Diffusion Equations." International Journal of Nonlinear Sciences and Numerical Simulation 19, no. 7-8 (December 19, 2018): 781–92. http://dx.doi.org/10.1515/ijnsns-2018-0111.

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Abstract:
AbstractIn this work, shifted fractional-order Jacobi orthogonal function in the interval $[0,\mathcal{T}]$ is outputted of the classical Jacobi polynomial (see Definition 2.3). Also, we list and derive some facts related to the shifted fractional-order Jacobi orthogonal function. Spectral collocation techniques are addressed to solve the multidimensional distributed-order diffusion equations (MDODEs). A mixed of shifted Jacobi polynomials and shifted fractional order Jacobi orthogonal functions are used as basis functions to adapt the spatial and temporal discretizations, respectively. Based on the selected basis, a spectral collocation method is listed to approximate the MDODEs. By means of the selected basis functions, the given conditions are automatically satisfied. We conclude with the application of spectral collocation method for multi-dimensional distributed-order diffusion equations.
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