×

Indefinite integrals of special functions from hybrid equations. (English) Zbl 1450.34003

Summary: Elementary linear first and second order differential equations can always be constructed for twice differentiable functions by explicitly including the function’s derivatives in the definition of these equations. If the function also obeys a conventional differential equation, information from this equation can be introduced into the elementary equations to give blended linear equations which are here called hybrid equations. Integration theorems are derived for these hybrid equations and several universal integrals are also derived. The paper presents integrals derived with these methods for cylinder functions, associated Legendre functions, and the Gegenbauer, Chebyshev, Hermite, Jacobi and Laguerre orthogonal polynomials. All the results presented have been checked using Mathematica.

MSC:

34A05 Explicit solutions, first integrals of ordinary differential equations
34A30 Linear ordinary differential equations and systems
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
33C55 Spherical harmonics

Software:

Mathematica
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Conway, JT., A Lagrangian method for deriving new indefinite integrals of special functions, Integral Transforms Spec Funct, 26, 10, 812-824 (2015) · Zbl 1372.70046 · doi:10.1080/10652469.2015.1052807
[2] Conway, JT., Indefinite integrals of some special functions from a new method, Integral Transforms Spec Funct, 26, 11, 845-858 (2015) · Zbl 1372.70047 · doi:10.1080/10652469.2015.1063627
[3] Conway, JT., Indefinite integrals involving the incomplete elliptic integral of the third kind, Integral Transforms Spec Funct, 27, 8, 667-682 (2016) · Zbl 1398.33009 · doi:10.1080/10652469.2016.1184662
[4] Conway, JT., Indefinite integrals involving the incomplete elliptic integrals of the first and second kinds, Integral Transforms Spec Funct, 27, 5, 371-384 (2016) · Zbl 1398.33008 · doi:10.1080/10652469.2015.1132715
[5] Conway, JT., Indefinite integrals of quotients of special functions, Integral Transforms Spec Funct, 29, 4, 269-283 (2018) · doi:10.1080/10652469.2018.1428582
[6] Conway, JT., Indefinite integrals of quotients of Gauss hypergeometric functions, Integral Transforms Spec Funct, 29, 6, 417-430 (2018) · Zbl 1387.33008 · doi:10.1080/10652469.2018.1451527
[7] Abel, NH., Oeuvres complètes (1839), Christiania: Grondahl, Christiania
[8] King, AC; Billingham, J.; Otto, SR., Differential equations, linear, nonlinear, partial (2003), Cambridge: Cambridge University Press, Cambridge · Zbl 1034.34001
[9] Prudnikov, AP; Brychkov, YuA; Marichev, OI., Integrals and series, Vol. 2, special functions (1986), New York (NY): Gordon and Breach, New York (NY) · Zbl 0733.00004
[10] Brychkov, YuA., Handbook of special functions: derivatives, integrals, series and other formulas (2008), Boca Raton (FL): Chapman & Hall/CRC, Boca Raton (FL) · Zbl 1158.33001
[11] Conway, JT., New special function recurrences giving new indefinite integrals, Integral Transforms Spec Funct, 29, 10, 805-819 (2018) · Zbl 1396.33008 · doi:10.1080/10652469.2018.1499099
[12] Gradshteyn, IS; Ryzhik, IM., Table of integrals, series and products (2007), New York (NY): Academic, New York (NY) · Zbl 1208.65001
[13] Conway, JT., Indefinite integrals of special functions from inhomogeneous differential equations, Integral Transforms Spec Funct, 30, 3, 166-180 (2018) · Zbl 1405.33003 · doi:10.1080/10652469.2018.1548014
[14] Wolfram, S., The mathematica book (2003), Champaign (IL): Wolfram Media, Champaign (IL)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.