# Ferrers board

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##### 1: 14.1 Special Notation

###### §14.1 Special Notation

… ►The main functions treated in this chapter are the Legendre functions ${\U0001d5af}_{\nu}\left(x\right)$, ${\U0001d5b0}_{\nu}\left(x\right)$, ${P}_{\nu}\left(z\right)$, ${Q}_{\nu}\left(z\right)$; Ferrers functions ${\U0001d5af}_{\nu}^{\mu}\left(x\right)$, ${\U0001d5b0}_{\nu}^{\mu}\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions ${P}_{\nu}^{\mu}\left(z\right)$, ${Q}_{\nu}^{\mu}\left(z\right)$, ${\bm{Q}}_{\nu}^{\mu}\left(z\right)$; conical functions ${\U0001d5af}_{-\frac{1}{2}+\mathrm{i}\tau}^{\mu}\left(x\right)$, ${\U0001d5b0}_{-\frac{1}{2}+\mathrm{i}\tau}^{\mu}\left(x\right)$, ${\widehat{\U0001d5b0}}_{-\frac{1}{2}+\mathrm{i}\tau}^{\mu}\left(x\right)$, ${P}_{-\frac{1}{2}+\mathrm{i}\tau}^{\mu}\left(x\right)$, ${Q}_{-\frac{1}{2}+\mathrm{i}\tau}^{\mu}\left(x\right)$ (also known as Mehler functions). ►Among other notations commonly used in the literature Erdélyi et al. (1953a) and Olver (1997b) denote ${\U0001d5af}_{\nu}^{\mu}\left(x\right)$ and ${\U0001d5b0}_{\nu}^{\mu}\left(x\right)$ by ${\mathrm{P}}_{\nu}^{\mu}(x)$ and ${\mathrm{Q}}_{\nu}^{\mu}(x)$, respectively. …Hobson (1931) denotes both ${\U0001d5af}_{\nu}^{\mu}\left(x\right)$ and ${P}_{\nu}^{\mu}\left(x\right)$ by ${P}_{\nu}^{\mu}\left(x\right)$; similarly for ${\U0001d5b0}_{\nu}^{\mu}\left(x\right)$ and ${Q}_{\nu}^{\mu}\left(x\right)$.##### 2: 26.15 Permutations: Matrix Notation

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###### Example 2

►The*Ferrers board*of shape $({b}_{1},{b}_{2},\mathrm{\dots},{b}_{n})$, $0\le {b}_{1}\le {b}_{2}\le \mathrm{\cdots}\le {b}_{n}$, is the set $B=\{(j,k)|\mathrm{\hspace{0.25em}1}\le j\le n,1\le k\le {b}_{j}\}$. …If $B$ is the Ferrers board of shape $(0,1,2,\mathrm{\dots},n-1)$, then …##### 3: Christopher J. Howls

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►In 2008 he was appointed to the editorial board of the Proceedings of the Royal Society of London Series A.
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##### 4: Mourad E. H. Ismail

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►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*. …##### 5: Ronald F. Boisvert

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►Boisvert has served as Editor-in-Chief of the ACM Transactions on Mathematical Software 1992-2005, Co-chair of the Numerics Working Group of the Java Grande Forum 1998-2003, Co-chair of the Publications Board of the Association for Computing Machinery (ACM) 2005-2013, and Chair of the International Federation for Information Processing (IFIP) Working Group 2.
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##### 6: Peter A. Clarkson

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►He is a member of the editorial boards of nine international journals and has served as Chair, Vice-Chair, and Secretary of the SIAM Activity Group on Orthogonal Polynomials and Special Functions.
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##### 7: Tom H. Koornwinder

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►Currently he is on the editorial board for Constructive Approximation, and is editor for the volume on

*Multivariable Special Functions*in the ongoing Askey–Bateman book project. …##### 8: Peter Paule

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►He is on the editorial boards for the

*Journal of Symbolic Computation*and*The Ramanujan Journal*, and is Managing Editor of*Annals of Combinatorics*. …##### 9: Donald St. P. Richards

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►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.
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##### 10: Wadim Zudilin

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►He is a member of several editorial boards including the series Monographs in Number Theory published by World Scientific.
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