second order differential operators

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2: DLMF Project News
error generating summary
3: 18.3 Definitions
• 1.

As eigenfunctions of second order differential operators (Bochner’s theorem, Bochner (1929)). See the differential equations $A(x)p_{n}^{\prime\prime}(x)+B(x)p_{n}^{\prime}(x)+\lambda_{n}p_{n}(x)=0$, in Table 18.8.1.

• 4: 18.39 Applications in the Physical Sciences
The nature of, and notations and common vocabulary for, the eigenvalues and eigenfunctions of self-adjoint second order differential operators is overviewed in §1.18. … The fundamental quantum Schrödinger operator, also called the Hamiltonian, $\mathcal{H}$, is a second order differential operator of the form …
5: Errata
This especially included updated information on matrix analysis, measure theory, spectral analysis, and a new section on linear second order differential operators and eigenfunction expansions. … The specific updates to Chapter 1 include the addition of an entirely new subsection §1.18 entitled “Linear Second Order Differential Operators and Eigenfunction Expansions” which is a survey of the formal spectral analysis of second order differential operators. …

• 6: 19.4 Derivatives and Differential Equations
§19.4(ii) Differential Equations
Then …If $\phi=\pi/2$, then these two equations become hypergeometric differential equations (15.10.1) for $K\left(k\right)$ and $E\left(k\right)$. An analogous differential equation of third order for $\Pi\left(\phi,\alpha^{2},k\right)$ is given in Byrd and Friedman (1971, 118.03).
7: 2.9 Difference Equations
or equivalently the second-order homogeneous linear difference equation …in which $\Delta$ is the forward difference operator3.6(i)). … This situation is analogous to second-order homogeneous linear differential equations with an irregular singularity of rank 1 at infinity (§2.7(ii)). … For asymptotic approximations to solutions of second-order difference equations analogous to the Liouville–Green (WKBJ) approximation for differential equations (§2.7(iii)) see Spigler and Vianello (1992, 1997) and Spigler et al. (1999). … For discussions of turning points, transition points, and uniform asymptotic expansions for solutions of linear difference equations of the second order see Wang and Wong (2003, 2005). …
8: Bibliography T
• E. C. Titchmarsh (1946) Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press, Oxford.
• E. C. Titchmarsh (1958) Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations. Clarendon Press, Oxford.
• E. C. Titchmarsh (1962a) Eigenfunction expansions associated with second-order differential equations. Part I. Second edition, Clarendon Press, Oxford.
• S. Tsujimoto, L. Vinet, and A. Zhedanov (2012) Dunkl shift operators and Bannai-Ito polynomials. Adv. Math. 229 (4), pp. 2123–2158.
• S. A. Tumarkin (1959) Asymptotic solution of a linear nonhomogeneous second order differential equation with a transition point and its application to the computations of toroidal shells and propeller blades. J. Appl. Math. Mech. 23, pp. 1549–1565.
• 9: Bibliography D
• T. M. Dunster, D. A. Lutz, and R. Schäfke (1993) Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions. Proc. Roy. Soc. London Ser. A 440, pp. 37–54.
• T. M. Dunster (1990b) Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point. SIAM J. Math. Anal. 21 (6), pp. 1594–1618.
• T. M. Dunster (1994b) Uniform asymptotic solutions of second-order linear differential equations having a simple pole and a coalescing turning point in the complex plane. SIAM J. Math. Anal. 25 (2), pp. 322–353.
• T. M. Dunster (1996a) Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function. Proc. Roy. Soc. London Ser. A 452, pp. 1331–1349.
• A. J. Durán and F. A. Grünbaum (2005) A survey on orthogonal matrix polynomials satisfying second order differential equations. J. Comput. Appl. Math. 178 (1-2), pp. 169–190.
• 10: 2.8 Differential Equations with a Parameter
The transformed differential equation is …
§2.8(iv) Case III: Simple Pole
For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). … For further examples of uniform asymptotic approximations in terms of parabolic cylinder functions see §§13.20(iii), 13.20(iv), 14.15(v), 15.12(iii), 18.24. … Lastly, for an example of a fourth-order differential equation, see Wong and Zhang (2007). …