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1: 29.3 Definitions and Basic Properties

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Table 29.3.1: Eigenvalues of Lamé’s equation.

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eigenvalue $h$	parity	period
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Table 29.3.2: Lamé functions.

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boundary conditions	eigenvalue $h$	eigenfunction $w(z)$	parity of $w(z)$	parity of $w(z-K)$	period of $w(z)$
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2: 29.12 Definitions

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Table 29.12.1: Lamé polynomials.

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$\nu$	eigenvalue $h$	eigenfunction $w(z)$	polynomial form	real period	imag. period	parity of $w(z)$	parity of $w(z-K)$	parity of $w(z-K-\mathrm{i}{K}^{\prime })$
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3: 26.13 Permutations: Cycle Notation

… ►A permutation is even or odd according to the parity of the number of transpositions. …

4: 28.2 Definitions and Basic Properties

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§28.2(vi) Eigenfunctions

… ►Even parity means $w(-z)=w(z)$, and odd parity means $w(-z)=-w(z)$. … ►

Table 28.2.2: Eigenfunctions of Mathieu’s equation.

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Eigenvalues	Eigenfunctions	Periodicity	Parity
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5: 1.11 Zeros of Polynomials

… ►The number of positive zeros of a polynomial with real coefficients cannot exceed the number of times the coefficients change sign, and the two numbers have same parity. …

6: 14.30 Spherical and Spheroidal Harmonics

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Parity Operation

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7: 37.4 Disk with Weight Function ${\left(1-x^2-y^2\right)}^{\alpha }$

… ►The space ${𝒱}_n^{\gamma }(x,y)$ of OPs of $x,y$ of degree $n$ on $𝔻$ can be decomposed as a direct sum of two spaces of type ${𝒱}_m^{\alpha ,\beta ,\gamma }$ of OPs on $\mathrm{△}$ in a way which depends on the parity of $n$: …

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