https://en.formulasearchengine.com/w/index.php?action=history&feed=atom&title=Differential_dynamic_programmingDifferential dynamic programming - Revision history2026-08-05T10:38:23ZRevision history for this page on the wikiMediaWiki 1.47.0-wmf.7https://en.formulasearchengine.com/w/index.php?title=Differential_dynamic_programming&diff=27244&oldid=preven>Psiorx: /* Differential dynamic programming */2013-09-09T04:30:37Z<p><span class="autocomment">Differential dynamic programming</span></p>
<p><b>New page</b></p><div>{{lowercase|''p''-adic gamma function}}<br />
In mathematics, the '''''p''-adic gamma function''' Γ<sub>''p''</sub>(''s'') is a function of a [[p-adic]] variable ''s'' analogous to the [[gamma function]]. It was first explicitly defined by {{harvtxt|Morita|1975}}, though {{harvtxt|Boyarsky|1980}} pointed out that {{harvtxt|Dwork|1964}} implicitly used the same function. {{harvtxt|Diamond|1977}} defined a ''p''-adic analog ''G''<sub>''p''</sub>(''s'') of log Γ(''s''). {{harvtxt|Overholtzer|1952}} had previously given a definition of a different ''p''-adic analogue of the gamma function, but his function does not have satisfactory properties and is not used much.<br />
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==Definition==<br />
The ''p''-adic gamma function is the unique continuous function of a ''p''-adic integer ''s'' such that<br />
<br />
:<math>\Gamma_p(s)=(-1)^s\prod_{0<i<s,\ p\nmid i}i</math><br />
<br />
for positive integers ''s'', where the product is restricted to integers ''i'' not divisible by ''p''.<br />
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==See also==<br />
<br />
*[[Gross–Koblitz formula]]<br />
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==References==<br />
*{{Citation | last1=Boyarsky | first1=Maurizio | title=p-adic gamma functions and Dwork cohomology | url=http://dx.doi.org/10.2307/1998301 | doi=10.2307/1998301 | id={{MR|552263}} | year=1980 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=257 | issue=2 | pages=359–369}}<br />
*{{Citation | last1=Diamond | first1=Jack | title=The p-adic log gamma function and p-adic Euler constants | url=http://www.jstor.org/stable/1997840 | id={{MR|0498503}} | year=1977 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=233 | pages=321–337}}<br />
*{{Citation | last1=Diamond | first1=Jack | editor1-last=Chudnovsky | editor1-first=David V. | editor1-link=Chudnovsky brothers | editor2-last=Chudnovsky | editor2-first=Gregory V. | editor3-last=Cohn | editor3-first=Henry | editor4-last=Nathanson | editor4-first=Melvyn B. | title=Number theory (New York, 1982) | url=http://dx.doi.org/10.1007/BFb0071542 | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Math. | isbn=978-3-540-12909-7 | doi=10.1007/BFb0071542 | id={{MR|750664}} | year=1984 | volume=1052 | chapter=p-adic gamma functions and their applications | pages=168–175}}<br />
*{{Citation | last1=Dwork | first1=Bernard | title=On the zeta function of a hypersurface. II | url=http://www.jstor.org/stable/1970392 | id={{MR|0188215}} | year=1964 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=80 | pages=227–299}}<br />
*{{Citation | last1=Morita | first1=Yasuo | title=A p-adic analogue of the Γ-function | url=http://hdl.handle.net/2261/6494 | id={{MR|0424762}} | year=1975 | journal=Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics | issn=0040-8980 | volume=22 | issue=2 | pages=255–266}}<br />
*{{Citation | last1=Overholtzer | first1=Gordon | title=Sum functions in elementary p-adic analysis | url=http://www.jstor.org/stable/2371998 | id={{MR|0048493}} | year=1952 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=74 | pages=332–346}}<br />
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[[Category:Number theory]]</div>en>Psiorx