Abstract
Families of orthogonal polynomials play a very important role in numerous applications in mathematical physics, as part of the class of so-called special functions. This is a classical topic in mathematics and has been the subject of investigation for centuries now. Many of these special functions were first introduced into mathematics as solutions of specific second-order differential equations. Second-order ordinary differential equations comprise one of the important topics that I do not discuss in this book owing to limitations of space, given that you are very likely to be already familiar with the rudiments of this topic. In any case, in applications we are interested primarily in the properties of the solutions of these equations. Between this chapter and those on analytic functions of a complex variable, there will be a fair amount of discussion of these properties.
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Notes
- 1.
This is just one of the motivating reasons for the generalization. There are other reasons as well. For instance, orthogonal polynomials are related to group representations, and weight functions occur naturally in that context, as so-called invariant measures over group manifolds.
- 2.
It is customary to write \(\rho (x)\, dx \equiv d\mu (x)\) for the measure of integration. If the indefinite integral (or primitive) of the weight \(\rho \,(x)\) exists in explicit form, then we would have \(\mu (x) = \int \rho \,(x)\, dx\). But \(\mu (x)\) may not even exist as an elementary function, or even in explicit form, for a general \(\rho \,(x)\)—nor is it required to do so. For example, there is no elementary function \(\mu (x)\) such that \(d\mu (x)/dx = e^{-x^2}\). To keep things simple, I shall just write out the measure as \(\rho \,(x)\, dx\).
- 3.
But see the remarks that follow shortly.
- 4.
NU, p. 34: Eq. (6) of Chap. II, Sect. 6.2.
- 5.
NU, pp. 36 and 37: Eqs. (7) and (11) of Chap. II, Sect. 6.3.
- 6.
NU, p. 2: Eq. (9) of Chap. I, Sect. 1.
- 7.
We will discuss analytic continuation in Chap. 25.
- 8.
It is assumed that the reader is familiar with this test to determine the absolute convergence or otherwise of an infinite series. We will discuss the convergence of power series (including the ratio test) in a little more detail in Chap. 22, Sect. 22.5.
- 9.
Nikiforov and Uvarov, op. cit.
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Balakrishnan, V. (2020). Orthogonal Polynomials. In: Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-39680-0_16
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