contiguous%20relations

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21: 10.21 Zeros

… ►The positive zeros of any two real distinct cylinder functions of the same order are interlaced, as are the positive zeros of any real cylinder function ${𝒞}_{\nu }\left(z\right)$ and the contiguous function ${𝒞}_{\nu +1}\left(z\right)$. … … ►The functions ${\rho }_{\nu }(t)$ and ${\sigma }_{\nu }(t)$ are related to the inverses of the phase functions ${\theta }_{\nu }\left(x\right)$ and ${\varphi }_{\nu }\left(x\right)$ defined in §10.18(i): if $\nu \ge 0$, then … ►

${\varphi }_{\nu }\left(y_{\nu ,m}^{\prime }\right)=m\pi ,$ $m=1,2,\mathrm{\dots }$.

…

22: 27.2 Functions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►

Table 27.2.2: Functions related to division.

► ►►►►►

$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
6	2	4	12	19	18	2	20	32	16	6	63	45	24	6	78
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…

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23: 26.6 Other Lattice Path Numbers

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Delannoy Number $D(m,n)$

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Motzkin Number $M(n)$

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Narayana Number $N(n,k)$

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Schröder Number $r(n)$

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§26.6(iii) Recurrence Relations

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24: 16.7 Relations to Other Functions

§16.7 Relations to Other Functions

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25: 6.16 Mathematical Applications

… ►If we assume Riemann’s hypothesis that all nonreal zeros of $\zeta \left(s\right)$ have real part of $\frac{1}{2}$ (§25.10(i)), then … ►

►►

Figure 6.16.2: The logarithmic integral $\mathrm{li}\left(x\right)$, together with vertical bars indicating the value of $\pi (x)$ for $x=10,20,\mathrm{\dots },1000$. Magnify

26: 26.9 Integer Partitions: Restricted Number and Part Size

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Table 26.9.1: Partitions $p_k\left(n\right)$.

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$n$	$k$
	…
8	0	1	5	10	15	18	20	21	22	22	22
…

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… ►It follows that $p_k\left(n\right)$ also equals the number of partitions of $n$ into parts that are less than or equal to $k$. … ►

§26.9(iii) Recurrence Relations

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27: 6.11 Relations to Other Functions

§6.11 Relations to Other Functions

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Incomplete Gamma Function

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Confluent Hypergeometric Function

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6.11.2 $$E_1\left(z\right)={\mathrm{e}}^{-z}U(1,1,z),$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $z$: complex variable

Referenced by:

§6.11, 5th item, §6.6

Permalink:

http://dlmf.nist.gov/6.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

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6.11.3 $$\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right)=U(1,1,-\mathrm{i}z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{i}$: imaginary unit, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.11, 5th item

Permalink:

http://dlmf.nist.gov/6.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

28: Publications

DLMF Related Publications

… ►

NIST’s Digital Library of Mathematial Functions, Physics Today 71, 2, 48 (2018), pp. 48–53.

29: 8.17 Incomplete Beta Functions

… ►

8.17.4 $$I_x(a,b)=1-I_{1-x}(b,a).$$

ⓘ

Symbols:

$I_x(a,b)$: incomplete beta function, $a$: parameter, $b$: parameter and $x$: variable

A&S Ref:

6.6.3 (Symmetry relation)

Referenced by:

§8.17(v), §8.18(i)

Permalink:

http://dlmf.nist.gov/8.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.17(i), §8.17 and Ch.8

►

8.17.5 $$I_x(m,n-m+1)=\sum _{j=m}^n\left(\genfrac{}{}{0.0pt}{}{n}{j}\right)x^j{(1-x)}^{n-j},$$ $m,n$ positive integers; .

ⓘ

Symbols:

$I_x(a,b)$: incomplete beta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer and $x$: variable

A&S Ref:

6.6.4 (Relation to the Binomial Expansion)

Referenced by:

§8.17(i), §8.17(vii), Erratum (V1.0.5) for Subsection 8.17(i)

Permalink:

http://dlmf.nist.gov/8.17.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.17(i), §8.17 and Ch.8

… ►

§8.17(ii) Hypergeometric Representations

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§8.17(iv) Recurrence Relations

… ►

8.17.24 $$I_x(m,n)={(1-x)}^n\sum _{j=m}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{n+j-1}{j}\right)x^j,$$ $m,n$ positive integers; .

ⓘ

Symbols:

$I_x(a,b)$: incomplete beta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer and $x$: variable

A&S Ref:

26.5.26 (The upper limit of summation has been corrected.)

Referenced by:

§8.17(i), Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/8.17.E24

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation is the same as Equation (26.5.26) of Abramowitz and Stegun (1964), except that the upper limit of the summation has been corrected to $\mathrm{\infty }$.

Suggested 2011-03-23 by Stephen Bourn

See also:

Annotations for §8.17(vii), §8.17 and Ch.8

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30: 25 Zeta and Related Functions

Chapter 25 Zeta and Related Functions

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