American stirlingses Reservation Number 📞818.273.1342�

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1: 24.1 Special Notation

… ►

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: Mourad E. H. Ismail

… ► 190, American Mathematical Society, 1995; Special Functions,

q

-Series and Related Topics (with D. … 14, American Mathematical Society, 1997;

q

-Series from a Contemporary Perspective (with D. … 254, American Mathematical Society, 2000; Special Functions—Proceedings of the International Workshop, Hong Kong, June 21–25, 1999, World Scientific, 2000; Special Functions 2000: Current Perspective and Future Directions (with J. … ►Ismail serves on several editorial boards including the Cambridge University Press book series Encyclopedia of Mathematics and its Applications, and on the editorial boards of 9 journals including Proceedings of the American Mathematical Society (Integrable Systems and Special Functions Editor); Constructive Approximation; Journal of Approximation Theory; and Integral Transforms and Special Functions. ►Ismail was elected a fellow of the American Mathematical Society in 2014. …

3: David M. Bressoud

… ►Bressoud has published numerous papers in number theory, combinatorics, and special functions. His books are Analytic and Combinatorial Generalizations of the Rogers-Ramanujan Identities, published in Memoirs of the American Mathematical Society 24, No. 227, in 1980, Factorization and Primality Testing, published by Springer-Verlag in 1989, Second Year Calculus from Celestial Mechanics to Special Relativity, published by Springer-Verlag in 1992, A Radical Approach to Real Analysis, published by the Mathematical Association of America in 1994, with a second edition in 2007, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, published by the Mathematical Association of America and Cambridge University Press in 1999, A Course in Computational Number Theory (with S. … Parson) of The Rademacher Legacy to Mathematics, published by the American Mathematical Society in 1994. …

4: Peter L. Walker

… … ►was Professor of Mathematics at the American University of Sharjah, Sharjah, United Arab Emirates, in 1997–2005. … ►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. …

5: 26.8 Set Partitions: Stirling Numbers

§26.8 Set Partitions: Stirling Numbers

… ►

s\left(n,k\right)

denotes the Stirling number of the first kind:

(-1)^{n-k}

times the number of permutations of

\{1,2,\ldots,n\}

with exactly

k

cycles. … … ►Let

A

and

B

be the

n\times n

matrices with

(j,k)

th elements

s\left(j,k\right)

, and

S\left(j,k\right)

, respectively. … ►For asymptotic approximations for

s\left(n+1,k+1\right)

and

S\left(n,k\right)

that apply uniformly for

1\leq k\leq n

n\to\infty

see Temme (1993) and Temme (2015, Chapter 34). …

6: George E. Andrews

… ►An expert on

q

-series, he is the author of

q

-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. … ►Andrews was elected to the American Academy of Arts and Sciences in 1997, and to the National Academy of Sciences (USA) in 2003. …

7: 26.1 Special Notation

… ► ►►►►►

$\genfrac{(}{)}{0.0pt}{}{m}{n}$	binomial coefficient.
…
$B\left(n\right)$	Bell number.
$C\left(n\right)$	Catalan number.
…
$s\left(n,k\right)$	Stirling numbers of the first kind.
…

… ►Other notations for

s\left(n,k\right)

, the Stirling numbers of the first kind, include

S_{n}^{(k)}

(Abramowitz and Stegun (1964, Chapter 24), Fort (1948)),

S_{n}^{k}

(Jordan (1939), Moser and Wyman (1958a)),

\genfrac{(}{)}{0.0pt}{}{n-1}{k-1}B_{n-k}^{(n)}

(Milne-Thomson (1933)),

(-1)^{n-k}S_{1}(n-1,n-k)

(Carlitz (1960), Gould (1960)),

(-1)^{n-k}\left[n\atop k\right]

(Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). ►Other notations for

S\left(n,k\right)

, the Stirling numbers of the second kind, include

\mathscr{S}^{(k)}_{n}

(Fort (1948)),

\mathfrak{S}_{n}^{k}

(Jordan (1939)),

\sigma_{n}^{k}

(Moser and Wyman (1958b)),

\genfrac{(}{)}{0.0pt}{}{n}{k}B_{n-k}^{(-k)}

(Milne-Thomson (1933)),

S_{2}(k,n-k)

(Carlitz (1960), Gould (1960)),

\left\{n\atop k\right\}

(Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).

8: Ira Gessel

… ►He is a Fellow of the American Mathematical Society. …

9: Donald St. P. Richards

… ►He is editor of the book Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, published by the American Mathematical Society in 1992, and coeditor of Representation Theory and Harmonic Analysis: A Conference in Honor of R. A. Kunze (with T. … 191, by the American Mathematical Society in 1995. … ►He is also Associate Editor of the Annals of Statistics, Associate Review Editor of the Journal of the American Statistical Association, and Member of the Board of Governors of the Institute for Mathematics and its Applications. …

10: Barry I. Schneider

… ►Schneider’s current research interests span a broad number of areas of theoretical chemistry, atomic and molecular physics, numerical methods and high performance computing. …Schneider was awarded a Poste Rouge by the CNRS in 1980, was elected a Fellow of the American Physical Society (APS) in 1983 and received the prestigious Humboldt prize from the German government in 1987. …Schneider has served as Chair and Co-Chair of the APS Division of Computational Physics and the Topical Group on Few-Body Systems and Multipartical Dynamics and has been the organizer of a number of conferences and invited sessions here and abroad. …