Ince theorem

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1: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

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§28.5(i) Definitions

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Theorem of Ince (1922)

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2: 28.2 Definitions and Basic Properties

… ►If $q\ne 0$, then for a given value of $\nu$ the corresponding Floquet solution is unique, except for an arbitrary constant factor (Theorem of Ince; see also 28.5(i)). …

3: Errata

… ►We now include Markov’s Theorem. In regard to orthogonal polynomials on the unit circle, we now discuss monic polynomials, Verblunsky’s Theorem, and Szegő’s theorem. … ►

The largest known prime, which is a Mersenne prime, was updated from $2^{43,112,609}-1$ (2009) to $2^{82,589,933}-1$ (2018).

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In Equation (1.13.4), the determinant form of the two-argument Wronskian

1.13.4

was added as an equality. In ¶Wronskian (in §1.13(i)), immediately below Equation (1.13.4), a sentence was added indicating that in general the $n$-argument Wronskian is given by $𝒲\left\{w_1(z),\mathrm{\dots },w_n(z)\right\}=det\left[w_k^{(j-1)}(z)\right]$, where $1\le j,k\le n$. Immediately below Equation (1.13.4), a sentence was added giving the definition of the $n$-argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for $n$th-order differential equations. A reference to Ince (1926, §5.2) was added.

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33.14.15 $${\int }_0^{\mathrm{\infty }}{\varphi }_{m,\mathrm{\ell }}(r){\varphi }_{n,\mathrm{\ell }}(r)dr={\delta }_{m,n}$$

The definite integral, originally written as ${\int }_0^{\mathrm{\infty }}{\varphi }_{n,\mathrm{\ell }}^2(r)dr=1$, was clarified and rewritten as an orthogonality relation. This follows from (33.14.14) by combining it with Dunkl (2003, Theorem 2.2).

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