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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: Bibliography O

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Studies on the Painlevé equations. II. Fifth Painlevé equation $P_{\mathrm{V}}$ . Japan. J. Math. (N.S.) 13 (1), pp. 47–76.

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External Links:

ISSN 0289-2316, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.10(v), §32.6(v), §32.7(viii).

Encodings:

BibTeX

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Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.

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External Links:

ISSN 0219-5305, Document, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

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Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.

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External Links:

Document, MathReview (Anthony Leung), ZentralBlatt

Referenced by:

§2.8(v), (9.13.13), (9.13.17), §9.13(i), §9.13(i).

Encodings:

BibTeX

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Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.

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External Links:

Document, MathReview (W. Wasow), ZentralBlatt

Referenced by:

§2.8(v).

Encodings:

BibTeX

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Formulae and Tables, The Modified Quotients of Cylinder Functions. Technical report Technical Report UDC 517.564.3:518.25, Vol. 4, Report of the Institute of Industrial Science, University of Tokyo, Institute of Industrial Science, Chiba City, Japan.

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Referenced by:

§10.29(i), §10.6(i).

Encodings:

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3: Bibliography N

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Toda equation and its solutions in special functions. J. Phys. Soc. Japan 65 (6), pp. 1589–1597.

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External Links:

ISSN 0031-9015, Document, MathReview, ZentralBlatt

Referenced by:

§18.38(ii).

Encodings:

BibTeX

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The Development of Prime Number Theory: From Euclid to Hardy and Littlewood. Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-66289-8, MathReview (D. R. Heath-Brown), ZentralBlatt

Referenced by:

§27.12.

Encodings:

BibTeX

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Generalization of Binet’s Gamma function formulas. Integral Transforms Spec. Funct. 24 (8), pp. 597–606.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview (Daniele Ritelli), ZentralBlatt

Referenced by:

§5.11(i), §5.11(i).

Encodings:

BibTeX

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An Introduction to the Theory of Numbers. 5th edition, John Wiley & Sons Inc., New York.

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External Links:

ISBN 0-471-62546-9, MathReview, ZentralBlatt

Referenced by:

Ch.27.

Encodings:

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Notes:

References and links to software for factorization and primality testing.

External Links:

Home Page

Referenced by:

6th item.

Encodings:

BibTeX

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4: Bibliography Y

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Computation of Kummer functions $U(a,b,x)$ for large argument $x$ by using the $\tau$-method. Trans. Inform. Process. Soc. Japan 36 (10), pp. 2335–2342 (Japanese).

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Notes:

Japanese with English summary. Double-precision Fortran.

External Links:

ISSN 0387-5806, FullText, MathReview

Referenced by:

§13.32(ii).

Encodings:

BibTeX

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5: Guide to Searching the DLMF

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a textual word, a number, or a math symbol.

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any double-quoted sequence of textual words and numbers.

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adj, prec/n, and near/n, where n is any positive natural number.

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$	stands for any number of alphanumeric characters
	(the more conventional * is reserved for the multiplication operator)
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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: 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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8: 24.15 Related Sequences of Numbers

§24.15 Related Sequences of Numbers

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§24.15(i) Genocchi Numbers

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§24.15(ii) Tangent Numbers

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§24.15(iii) Stirling Numbers

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§24.15(iv) Fibonacci and Lucas Numbers

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9: 26.13 Permutations: Cycle Notation

… ►is $\left(1,3,2,5,7\right)\left(4\right)\left(6,8\right)$ in cycle notation. …In consequence, (26.13.2) can also be written as $\left(1,3,2,5,7\right)\left(6,8\right)$. … ►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 … ►The derangement number, $d(n)$, is the number of elements of ${𝔖}_n$ with no fixed points: …

10: 27.2 Functions

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§27.2(i) Definitions

… ►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 $n$. … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ► … ►

§27.2(ii) Tables

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