Liouville function

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1: 27.2 Functions

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27.2.13

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Defines:

$\lambda \left(n\right)$: Liouville’s function

Symbols:

$\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer and $a$: primitive root

Permalink:

http://dlmf.nist.gov/27.2.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

►This is Liouville’s function. …

2: Bibliography L

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Higher monotonicity properties of certain Sturm-Liouville functions. III. Canad. J. Math. 22, pp. 1238–1265.

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External Links:

ISSN 0008-414X, MathReview (G. Sansone), ZentralBlatt

Referenced by:

§10.21(ii).

Encodings:

BibTeX

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Higher monotonicity properties of certain Sturm-Liouville functions. IV. Canad. J. Math. 24, pp. 349–368.

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External Links:

ISSN 0008-414X, MathReview (G. Sansone), ZentralBlatt

Referenced by:

§10.21(ii).

Encodings:

BibTeX

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Higher monotonicity properties of certain Sturm-Liouville functions.. Acta Math. 109, pp. 55–73.

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External Links:

ISSN 0001-5962, Document, MathReview (G. Sansone), ZentralBlatt

Referenced by:

§10.21(ii).

Encodings:

BibTeX

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3: 27.6 Divisor Sums

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27.6.1

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Symbols:

$\lambda \left(n\right)$: Liouville’s function, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

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4: 27.7 Lambert Series as Generating Functions

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27.7.6 $$\sum _{n=1}^{\mathrm{\infty }}\lambda \left(n\right)\frac{x^n}{1-x^n}=\sum _{n=1}^{\mathrm{\infty }}{x}^{n^2}.$$

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Symbols:

$\lambda \left(n\right)$: Liouville’s function, $n$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.7 and Ch.27

5: Bibliography M

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Singular integrals whose kernels involve certain Sturm-Liouville functions. I. J. Math. Mech. 19 (10), pp. 855–873.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§9.10(ii).

Encodings:

BibTeX

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Higher monotonicity properties of certain Sturm-Liouville functions. V. Proc. Roy. Soc. Edinburgh Sect. A 77 (1-2), pp. 23–37.

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External Links:

MathReview (G. Sansone), ZentralBlatt

Referenced by:

§10.21(ii), (9.11.4), §9.11(iii).

Encodings:

BibTeX

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6: 27.4 Euler Products and Dirichlet Series

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27.4.7 $$\sum _{n=1}^{\mathrm{\infty }}\lambda \left(n\right)n^{-s}=\frac{\zeta \left(2s\right)}{\zeta \left(s\right)},$$ $\mathrm{\Re }s>1$,

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Symbols:

$\lambda \left(n\right)$: Liouville’s function, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

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7: 1.9 Calculus of a Complex Variable

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Liouville’s Theorem

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8: 15.16 Products

§15.16 Products

… ►where $A_0=1$ and $A_s$, $s=1,2,\mathrm{\dots }$, are defined by the generating function ►

15.16.2 $${(1-z)}^{a+b-c}F(2a,2b;2c-1;z)=\sum _{s=0}^{\mathrm{\infty }}{A}_s{z}^s,$$ .

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $A_s$: coefficients

Permalink:

http://dlmf.nist.gov/15.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.16 and Ch.15

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Generalized Legendre’s Relation

… ►For further results of this kind, and also series of products of hypergeometric functions, see Erdélyi et al. (1953a, §2.5.2).

9: 18.39 Applications in the Physical Sciences

… ►An important, and perhaps unexpected, feature of the EOP’s is now pointed out by noting that for 1D Schrödinger operators, or equivalent Sturm-Liouville ODEs, having discrete spectra with $L^2$ eigenfunctions vanishing at the end points, in this case $\pm \mathrm{\infty }$ see Simon (2005c, Theorem 3.3, p. 35), such eigenfunctions satisfy the Sturm oscillation theorem. …

10: 1.13 Differential Equations

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Liouville Transformation

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§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms

… ►This is the Sturm-Liouville form of a second order differential equation, where ^′ denotes $\frac{d}{dx}$. … ►A regular Sturm-Liouville system will only have solutions for certain (real) values of $\lambda$, these are eigenvalues. … ►

Transformation to Liouville normal Form

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