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In mathematics, the ”’Pidduck polynomials”’ ”s”<sub>”n”</sub>(”x”) are polynomials introduced by {{harvs|txt|last=Pidduck|year1=1910|year2=1912}}<ref>{{Citation |last1=Pidduck |first1=F. B. |title=On the Propagation of a Disturbance in a Fluid under Gravity |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |volume=83 |issue=563 |pages=347–356 |year=1910 |url=https://zenodo.org/record/1432013 |publisher=The Royal Society |bibcode=1910RSPSA..83..347P |doi=10.1098/rspa.1910.0023 |issn=0950-1207 |jstor=92977 |doi-access=free}}</ref><ref>{{Citation |last1=Pidduck |first1=F. B. |title=The Wave-Problem of Cauchy and Poisson for Finite Depth and Slightly Compressible Fluid |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |volume=86 |issue=588 |pages=396–405 |year=1912 |publisher=The Royal Society |bibcode=1912RSPSA..86..396P |doi=10.1098/rspa.1912.0031 |issn=0950-1207 |jstor=93103 |doi-access=free}}</ref> given by the [[generating function]]
In mathematics, the ”’Pidduck polynomials”’ ”s”<sub>”n”</sub>(”x”) are polynomials introduced by {{harvs|txt|last=Pidduck|year1=1910|year2=1912}} given by the [[generating function]]
:<math>\displaystyle \sum_n \frac{s_n(x)}{n!}t^n = \left(\frac{1+t}{1-t}\right)^x(1-t)^{-1}</math>
:<math>\displaystyle \sum_n \frac{s_n(x)}{n!}t^n = \left(\frac{1+t}{1-t}\right)^x(1-t)^{-1}</math>
==References==
==References==
<references />
*{{Citation | last1=Boas | first1=Ralph P. | last2=Buck | first2=R. Creighton | title=Polynomial expansions of analytic functions | url=https://books.google.com/books?id=eihMuwkh4DsC | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. | mr=0094466 | year=1958 | volume=19| isbn=978-0-387-03123-1 }}
*{{Citation | last1=Boas | first1=Ralph P. | last2=Buck | first2=R. Creighton | title=Polynomial expansions of analytic functions | url=https://books.google.com/books?id=eihMuwkh4DsC | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. | mr=0094466 | year=1958 | volume=19| isbn=978-0-387-03123-1 }}
*{{Citation | last1=Pidduck | first1=F. B. | title=On the Propagation of a Disturbance in a Fluid under Gravity | jstor=92977 | publisher=The Royal Society | year=1910 | journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character | issn=0950-1207 | volume=83 | issue=563 | pages=347–356 | doi=10.1098/rspa.1910.0023| url=https://zenodo.org/record/1432013 | doi-access=free | bibcode=1910RSPSA..83..347P }}
*{{Citation | last1=Pidduck | first1=F. B. | title=The Wave-Problem of Cauchy and Poisson for Finite Depth and Slightly Compressible Fluid | jstor=93103 | publisher=The Royal Society | year=1912 | journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character | issn=0950-1207 | volume=86 | issue=588 | pages=396–405 | doi=10.1098/rspa.1912.0031| doi-access=free | bibcode=1912RSPSA..86..396P }}
*{{Citation | last1=Roman | first1=Steven | title=The umbral calculus | url=https://books.google.com/books?id=JpHjkhFLfpgC | publisher=Academic Press Inc. [Harcourt Brace Jovanovich Publishers] | location=London | series=Pure and Applied Mathematics | isbn=978-0-12-594380-2 | mr=741185 | year=1984 | volume=111}} Reprinted by Dover Publications, 2005
*{{Citation | last1=Roman | first1=Steven | title=The umbral calculus | url=https://books.google.com/books?id=JpHjkhFLfpgC | publisher=Academic Press Inc. [Harcourt Brace Jovanovich Publishers] | location=London | series=Pure and Applied Mathematics | isbn=978-0-12-594380-2 | mr=741185 | year=1984 | volume=111}} Reprinted by Dover Publications, 2005
In mathematics, the Pidduck polynomials sn(x) are polynomials introduced by Pidduck (1910, 1912)[1][2] given by the generating function
- ∑ n s n ( x ) n ! t n = ( 1 + t 1 − t ) x ( 1 − t ) − 1 {\displaystyle \displaystyle \sum _{n}{\frac {s_{n}(x)}{n!}}t^{n}=\left({\frac {1+t}{1-t}}\right)^{x}(1-t)^{-1}}
(Roman 1984, 4.4.3), (Boas & Buck 1958, p.38)
- ^ Pidduck, F. B. (1910), “On the Propagation of a Disturbance in a Fluid under Gravity”, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 83 (563), The Royal Society: 347–356, Bibcode:1910RSPSA..83..347P, doi:10.1098/rspa.1910.0023, ISSN 0950-1207, JSTOR 92977
- ^ Pidduck, F. B. (1912), “The Wave-Problem of Cauchy and Poisson for Finite Depth and Slightly Compressible Fluid”, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 86 (588), The Royal Society: 396–405, Bibcode:1912RSPSA..86..396P, doi:10.1098/rspa.1912.0031, ISSN 0950-1207, JSTOR 93103
- Boas, Ralph P.; Buck, R. Creighton (1958), Polynomial expansions of analytic functions, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge., vol. 19, Berlin, New York: Springer-Verlag, ISBN 978-0-387-03123-1, MR 0094466
- Roman, Steven (1984), The umbral calculus, Pure and Applied Mathematics, vol. 111, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-594380-2, MR 0741185 Reprinted by Dover Publications, 2005
