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variants of this functions
LaguerreL






Mathematica Notation

Traditional Notation









Polynomials > LaguerreL[n,lambda,z] > Summation > Infinite summation





http://functions.wolfram.com/05.08.23.0014.01









  


  










Input Form





Sum[((t (1 - t)^b E^((a z t)/(1 - t)))^k LaguerreL[k, c + b k, z (1 + a k)])/ ((m - c - b k - k) Pochhammer[(m - c + q - b k - k)/q, p]), {k, 0, Infinity}] == (((1 - t)^(m - c) E^((z t)/(t - 1)))/p!) Sum[(((-t)^k (1 - t)^(j q - k) Pochhammer[-p, j] Pochhammer[-m - j q, k])/ (k! j! (m - c + j q - b k - k))) Hypergeometric1F1[1, (c - m + b + 1 + b k + k - j q)/(b + 1), (z t (b + 1 + a m - a c + a j q))/((1 - t) (b + 1))], {j, 0, p}, {k, 0, m + q j}] /; \[LeftBracketingBar] t (1 - t)^b E^((a z t)/(1 - t)) \[RightBracketingBar] < 1 && Element[m, Integers] && m >= 0 && Element[p, Integers] && p >= 0 && Element[q, Integers] && q >= 0










Standard Form





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MathML Form







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</lowlimit> <uplimit> <apply> <plus /> <ci> m </ci> <apply> <times /> <ci> j </ci> <ci> q </ci> </apply> </apply> </uplimit> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> p </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> j </ci> <ci> q </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> <ci> j </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> j </ci> <ci> q </ci> </apply> </apply> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <factorial /> <ci> j </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <ci> k </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> m </ci> <apply> <times /> <ci> j </ci> <ci> q </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric1F1 </ci> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> k </ci> <ci> b </ci> </apply> <ci> b </ci> <ci> c </ci> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> j </ci> <ci> q </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> z </ci> <ci> t </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> c </ci> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> m </ci> </apply> <apply> <times /> <ci> a </ci> <ci> j </ci> <ci> q </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <apply> <abs /> <apply> <times /> <ci> t </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <ci> b </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> a </ci> <ci> z </ci> <ci> t </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> <apply> <in /> <ci> p </ci> <integers /> </apply> <apply> <in /> <ci> q </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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Date Added to functions.wolfram.com (modification date)





2002-12-18





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