# linearization formulas

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## 1—10 of 37 matching pages

##### 1: 18.18 Sums

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###### §18.18(v) Linearization Formulas

►###### Chebyshev

… ►###### Ultraspherical

… ►###### Hermite

… ►See Rahman (1981) for the linearization formula for Jacobi polynomials and Zeng (1992) for the linearization coefficients for Laguerre polynomials. …##### 2: 18.17 Integrals

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►Formulas (18.17.45) and (18.17.49) are integrated forms of the linearization formulas (18.18.22) and (18.18.23), respectively.
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##### 3: 15.12 Asymptotic Approximations

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►By combination of the foregoing results of this subsection with the linear transformations of §15.8(i) and the connection formulas of §15.10(ii), similar asymptotic approximations for $F(a+{e}_{1}\lambda ,b+{e}_{2}\lambda ;c+{e}_{3}\lambda ;z)$ can be obtained with ${e}_{j}=\pm 1$ or $0$, $j=1,2,3$.
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##### 4: Howard S. Cohl

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►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series.
Howard is the project leader for the NIST Digital Repository of Mathematical Formulae seeding and development projects.
In this regard, he has been exploring mathematical knowledge management and the digital expression of mostly unambiguous context-free full semantic information for mathematical formulae.

##### 5: Bibliography I

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The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of ${J}_{0}(z)-i{J}_{1}(z)$ and of Bessel functions ${J}_{m}(z)$ of any real order $m$
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Linear Algebra Appl. 194, pp. 35–70.
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The method of isomonodromic deformations and relation formulas for the second Painlevé transcendent.
Izv. Akad. Nauk SSSR Ser. Mat. 51 (4), pp. 878–892, 912 (Russian).
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Quasi-linear Stokes phenomenon for the second Painlevé transcendent.
Nonlinearity 16 (1), pp. 363–386.
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Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution.
J. Phys. A 31 (17), pp. 4073–4113.
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##### 6: 15.8 Transformations of Variable

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►The transformation formulas between two hypergeometric functions in Group 2, or two hypergeometric functions in Group 3, are the linear transformations (15.8.1).
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##### 7: Bibliography S

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Non-linear transformations of divergent and slowly convergent sequences.
J. Math. Phys. 34, pp. 1–42.
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Introduction to the Theory of Linear Spaces.
Martino, Mansfield Center, CT.
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Non-linear integral equations for Heun functions.
Proc. Edinburgh Math. Soc. (2) 16, pp. 281–289.
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Liouville-Green approximations for a class of linear oscillatory difference equations of the second order.
J. Comput. Appl. Math. 41 (1-2), pp. 105–116.
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The linear differential equation whose solutions are the products of solutions of two given differential equations.
J. Math. Anal. Appl. 98 (1), pp. 130–147.
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##### 8: 18.38 Mathematical Applications

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►Linear ordinary differential equations can be solved directly in series of Chebyshev polynomials (or other OP’s) by a method originated by Clenshaw (1957).
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►If the nodes in a quadrature formula with a positive weight function are chosen to be the zeros of the $n$th degree OP with the same weight function, and the interval of orthogonality is the same as the integration range, then the weights in the quadrature formula can be chosen in such a way that the formula is exact for all polynomials of degree not exceeding $2n-1$.
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►While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations.
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►This gives also new structures and results in the one-variable case, but the obtained nonsymmetric special functions can now usually be written as a linear combination of two known special functions.
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►See Koornwinder (2007a, (3.13), (4.9), (4.10)) for explicit formulas.
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##### 9: 3.3 Interpolation

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###### Linear Interpolation

… ►###### Three-Point Formula

… ►###### Four-Point Formula

… ►###### Five-Point Formula

… ►###### Six-Point Formula

…##### 10: Bibliography J

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Tables of Functions with Formulae and Curves.
4th edition, Dover Publications, New York.
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Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II.
Phys. D 2 (3), pp. 407–448.
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Asymptotic formulas for the zeros of the Meixner polynomials.
J. Approx. Theory 96 (2), pp. 281–300.
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Efficient implementation of the Hardy-Ramanujan-Rademacher formula.
LMS J. Comput. Math. 15, pp. 341–359.
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