q-Stirling%20numbers

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1: 24.1 Special Notation

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Bernoulli Numbers and Polynomials

►The origin of the notation $B_n$, $B_n\left(x\right)$, is not clear. … ►

Euler Numbers and Polynomials

… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations $E_n$, $E_n\left(x\right)$, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

2: 17.3 $q$-Elementary and $q$-Special Functions

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§17.3(iii) Bernoulli Polynomials; Euler and Stirling Numbers

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$q$-Euler Numbers

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$q$-Stirling Numbers

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17.3.9 $$a_{m,s}\left(q\right)=\frac{q^{-\left(\genfrac{}{}{0.0pt}{}{s}{2}\right)}{(1-q)}^s}{{(q;q)}_s}\sum _{j=0}^s{(-1)}^j{q}^{\left(\genfrac{}{}{0.0pt}{}{j}{2}\right)}{\left[\genfrac{}{}{0.0pt}{}{s}{j}\right]}_q\frac{{(1-q^{s-j})}^m}{{(1-q)}^m}.$$

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Defines:

$a_{m,s}\left(q\right)$: $q$-Stirling number

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $j$: nonnegative integer, $m$: nonnegative integer and $s$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.3.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(iii), §17.3(iii), §17.3 and Ch.17

… ►The $A_{m,s}\left(q\right)$ are always polynomials in $q$, and the $a_{m,s}\left(q\right)$ are polynomials in $q$ for $0\le s\le m$. …

3: 26.5 Lattice Paths: Catalan Numbers

§26.5 Lattice Paths: Catalan Numbers

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§26.5(i) Definitions

► $C\left(n\right)$ is the Catalan number. … ►

§26.5(ii) Generating Function

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§26.5(iii) Recurrence Relations

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4: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

… ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Their product $m$ has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …

5: 27.2 Functions

… ►where $p_1,p_2,\mathrm{\dots },p_{\nu \left(n\right)}$ are the distinct prime factors of $n$, each exponent $a_r$ is positive, and $\nu \left(n\right)$ is the number of distinct primes dividing $n$. …Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ► … ►

§27.2(ii) Tables

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6: 26.6 Other Lattice Path Numbers

§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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§26.6(iv) Identities

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7: 24.20 Tables

§24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of $N_n$ and $E_n$ for $n=20(2)60$ and $n=8(2)42$, respectively. …

8: 26.14 Permutations: Order Notation

… ►As an example, $35247816$ is an element of ${𝔖}_8.$ The inversion number is the number of pairs of elements for which the larger element precedes the smaller: … ► ►The Eulerian number, denoted , is the number of permutations in ${𝔖}_n$ with exactly $k$ descents. …The Eulerian number is equal to the number of permutations in ${𝔖}_n$ with exactly $k$ excedances. … ►

§26.14(iii) Identities

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9: 26.13 Permutations: Cycle Notation

… ►The Stirling cycle numbers of the first kind, denoted by $\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$, count the number of permutations of $\{1,2,\mathrm{\dots },n\}$ with exactly $k$ cycles. They are related to Stirling numbers of the first kind by …See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations. … ►The derangement number, $d(n)$, is the number of elements of ${𝔖}_n$ with no fixed points: … ►A permutation is even or odd according to the parity of the number of transpositions. …

10: Peter L. Walker

… ►Walker’s published work has been mainly in real and complex analysis, with excursions into analytic number theory and geometry, the latter in collaboration with Professor Mowaffaq Hajja of the University of Jordan. … ►

20 Theta Functions

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