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• W2054553457 abstract "While attempting to give extensions of the well-known Hille-Hardy formula for the generalized Laguerre polynomials <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-brace upper L Subscript n Baseline Superscript left-parenthesis alpha right-parenthesis Baseline left-parenthesis x right-parenthesis right-brace> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{ {L_n}^{(alpha )}(x)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defined by <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis 1 minus t right-parenthesis Superscript negative 1 minus alpha Baseline exp left-bracket minus StartFraction x t Over 1 minus t EndFraction right-bracket equals sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts upper L Subscript n Baseline Superscript left-parenthesis alpha right-parenthesis Baseline left-parenthesis x right-parenthesis t Superscript n> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mi>t</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mi>α<!-- α --></mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:mi>exp</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mrow> <mml:mo>[</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:mfrac> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>x</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:mfrac> </mml:mrow> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:munderover> <mml:mo movablelimits=false>∑<!-- ∑ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:munderover> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>t</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{(1 - t)^{ - 1 - alpha }}exp left [ { - frac {{xt}} {{1 - t}}} right ] = sum limits _{n = 0}^infty {{L_n}^{(alpha )}} (x){t^n}</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> , the author applies here certain operational techniques and the method of finite mathematical induction to derive several bilinear generating functions associated with various classes of generalized hypergeometric polynomials. It is observed that the earlier works of Brafman [2], [3], [4], Chaundy [5], Meixner [12], Weisner [16], and others quoted in the literature, are only specialized or limiting forms of the results presented here." @default.
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• W2054553457 date "1969-01-01" @default.
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• W2054553457 title "An extension of the Hille-Hardy formula" @default.
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• W2054553457 doi "https://doi.org/10.1090/s0025-5718-1969-0243132-4" @default.
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