the laguerre-polya class´lgrabarek/lg/arcs.pdfedmond laguerre and george polya´ paul turan´...
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![Page 1: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/1.jpg)
Historical BackgroundCurrent Research
The Laguerre-Polya ClassNon-linear operators and the Riemann Hypothesis
Lukasz Grabarek
Department of MathematicsUniversity of Hawai
s
i at Manoa
April 23, 2010
Lukasz Grabarek The Laguerre-Polya Class
![Page 2: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/2.jpg)
Historical BackgroundCurrent Research
Bernhard RiemannEdmond Laguerre and George PolyaPaul TuranGeorge Csordas, Richard Varga, Timothy Norfolk
The Riemann Hypothesis (1859)The most famous of all unresolved problems.
ξ(x/2) = 8∫ ∞
0Φ(t) cos xt dt ,
Φ(t) =∞∑
n=1
(2n4π2e9t − 3n2πe5t
)e−n2πe4t
.
”...es ist sehr wahrscheinlich, daß alle Wurzeln reell sind.”
TheoremThe Riemann Hypothesis holds if and only if ξ belongs to theLaguerre-Polya class.
Lukasz Grabarek The Laguerre-Polya Class
![Page 3: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/3.jpg)
Historical BackgroundCurrent Research
Bernhard RiemannEdmond Laguerre and George PolyaPaul TuranGeorge Csordas, Richard Varga, Timothy Norfolk
The Riemann Hypothesis (1859)The most famous of all unresolved problems.
ξ(x/2) = 8∫ ∞
0Φ(t) cos xt dt ,
Φ(t) =∞∑
n=1
(2n4π2e9t − 3n2πe5t
)e−n2πe4t
.
”...es ist sehr wahrscheinlich, daß alle Wurzeln reell sind.”
TheoremThe Riemann Hypothesis holds if and only if ξ belongs to theLaguerre-Polya class.
Lukasz Grabarek The Laguerre-Polya Class
![Page 4: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/4.jpg)
Historical BackgroundCurrent Research
Bernhard RiemannEdmond Laguerre and George PolyaPaul TuranGeorge Csordas, Richard Varga, Timothy Norfolk
The Riemann Hypothesis (1859)The most famous of all unresolved problems.
ξ(x/2) = 8∫ ∞
0Φ(t) cos xt dt ,
Φ(t) =∞∑
n=1
(2n4π2e9t − 3n2πe5t
)e−n2πe4t
.
”...es ist sehr wahrscheinlich, daß alle Wurzeln reell sind.”
TheoremThe Riemann Hypothesis holds if and only if ξ belongs to theLaguerre-Polya class.
Lukasz Grabarek The Laguerre-Polya Class
![Page 5: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/5.jpg)
Historical BackgroundCurrent Research
Bernhard RiemannEdmond Laguerre and George PolyaPaul TuranGeorge Csordas, Richard Varga, Timothy Norfolk
The Riemann Hypothesis (1859)The most famous of all unresolved problems.
ξ(x/2) = 8∫ ∞
0Φ(t) cos xt dt ,
Φ(t) =∞∑
n=1
(2n4π2e9t − 3n2πe5t
)e−n2πe4t
.
”...es ist sehr wahrscheinlich, daß alle Wurzeln reell sind.”
TheoremThe Riemann Hypothesis holds if and only if ξ belongs to theLaguerre-Polya class.
Lukasz Grabarek The Laguerre-Polya Class
![Page 6: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/6.jpg)
Historical BackgroundCurrent Research
Bernhard RiemannEdmond Laguerre and George PolyaPaul TuranGeorge Csordas, Richard Varga, Timothy Norfolk
The Riemann Hypothesis (1859)The most famous of all unresolved problems.
ξ(x/2) = 8∫ ∞
0Φ(t) cos xt dt ,
Φ(t) =∞∑
n=1
(2n4π2e9t − 3n2πe5t
)e−n2πe4t
.
”...es ist sehr wahrscheinlich, daß alle Wurzeln reell sind.”
TheoremThe Riemann Hypothesis holds if and only if ξ belongs to theLaguerre-Polya class.
Lukasz Grabarek The Laguerre-Polya Class
![Page 7: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/7.jpg)
Historical BackgroundCurrent Research
Bernhard RiemannEdmond Laguerre and George PolyaPaul TuranGeorge Csordas, Richard Varga, Timothy Norfolk
The Laguerre-Polya Class (1914)
Functions in the Laguerre-Polya class areuniform limits of polynomials all of whose zerosare real...
...and only the functions in the Laguerre-Polyaclass enjoy this property.
Lukasz Grabarek The Laguerre-Polya Class
![Page 8: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/8.jpg)
Historical BackgroundCurrent Research
Bernhard RiemannEdmond Laguerre and George PolyaPaul TuranGeorge Csordas, Richard Varga, Timothy Norfolk
The Laguerre-Polya Class (1914)
Functions in the Laguerre-Polya class areuniform limits of polynomials all of whose zerosare real...
...and only the functions in the Laguerre-Polyaclass enjoy this property.
Lukasz Grabarek The Laguerre-Polya Class
![Page 9: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/9.jpg)
Historical BackgroundCurrent Research
Bernhard RiemannEdmond Laguerre and George PolyaPaul TuranGeorge Csordas, Richard Varga, Timothy Norfolk
The Laguerre-Polya Class (1914)
Functions in the Laguerre-Polya class areuniform limits of polynomials all of whose zerosare real...
...and only the functions in the Laguerre-Polyaclass enjoy this property.
Lukasz Grabarek The Laguerre-Polya Class
![Page 10: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/10.jpg)
Historical BackgroundCurrent Research
Bernhard RiemannEdmond Laguerre and George PolyaPaul TuranGeorge Csordas, Richard Varga, Timothy Norfolk
The Laguerre-Polya Class (1914)
Functions in the Laguerre-Polya class areuniform limits of polynomials all of whose zerosare real...
...and only the functions in the Laguerre-Polyaclass enjoy this property.
Lukasz Grabarek The Laguerre-Polya Class
![Page 11: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/11.jpg)
Historical BackgroundCurrent Research
Bernhard RiemannEdmond Laguerre and George PolyaPaul TuranGeorge Csordas, Richard Varga, Timothy Norfolk
The Turan Inequalities (1948)A necessary condition.
Theorem
If ϕ(x) =∑∞
k=0γkk! x
k is a function in theLaguerre-Polya class, then γ2
k − γk−1γk+1 ≥ 0 fork = 1,2,3, . . . .
Lukasz Grabarek The Laguerre-Polya Class
![Page 12: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/12.jpg)
Historical BackgroundCurrent Research
Bernhard RiemannEdmond Laguerre and George PolyaPaul TuranGeorge Csordas, Richard Varga, Timothy Norfolk
The Turan Inequalities (1948)A necessary condition.
Theorem
If ϕ(x) =∑∞
k=0γkk! x
k is a function in theLaguerre-Polya class, then γ2
k − γk−1γk+1 ≥ 0 fork = 1,2,3, . . . .
Lukasz Grabarek The Laguerre-Polya Class
![Page 13: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/13.jpg)
Historical BackgroundCurrent Research
Bernhard RiemannEdmond Laguerre and George PolyaPaul TuranGeorge Csordas, Richard Varga, Timothy Norfolk
The Riemann ξ Function
ξ(x/2) = 8∫ ∞
0Φ(t) cos xt dt ,
Φ(t) =∞∑
n=1
(2n4π2e9t − 3n2πe5t
)e−n2πe4t
.
Theorem (Csordas, Varga, Norfolk (1986))
The coefficients of the Riemann ξ function satisfy the Turaninequalities.
Lukasz Grabarek The Laguerre-Polya Class
![Page 14: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/14.jpg)
Historical BackgroundCurrent Research
Non-linear operators.
Example
Construct a degree 5 polynomial p(x) with zeros x = −1.
p(x)
= (x + 1)(x + 1)(x + 1)(x + 1)(x + 1)= 1 + 5x + 10x2 + 10x3 + 5x4 + x5
= a0 + a1x + a2x2 + a3x3 + a4x5 + a5x5 .
Replace ak with 3a2k − 4ak−1ak+1 + ak−2ak+2.
p(x) becomes 3 + 35x + 105x2 + 105x3 + 35x4 + 3x5....and the zeros remain real and negative.
Lukasz Grabarek The Laguerre-Polya Class
![Page 15: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/15.jpg)
Historical BackgroundCurrent Research
Non-linear operators.
Example
Construct a degree 5 polynomial p(x) with zeros x = −1.
p(x) = (x + 1)(x + 1)(x + 1)(x + 1)(x + 1)
= 1 + 5x + 10x2 + 10x3 + 5x4 + x5
= a0 + a1x + a2x2 + a3x3 + a4x5 + a5x5 .
Replace ak with 3a2k − 4ak−1ak+1 + ak−2ak+2.
p(x) becomes 3 + 35x + 105x2 + 105x3 + 35x4 + 3x5....and the zeros remain real and negative.
Lukasz Grabarek The Laguerre-Polya Class
![Page 16: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/16.jpg)
Historical BackgroundCurrent Research
Non-linear operators.
Example
Construct a degree 5 polynomial p(x) with zeros x = −1.
p(x) = (x + 1)(x + 1)(x + 1)(x + 1)(x + 1)= 1 + 5x + 10x2 + 10x3 + 5x4 + x5
= a0 + a1x + a2x2 + a3x3 + a4x5 + a5x5 .
Replace ak with 3a2k − 4ak−1ak+1 + ak−2ak+2.
p(x) becomes 3 + 35x + 105x2 + 105x3 + 35x4 + 3x5....and the zeros remain real and negative.
Lukasz Grabarek The Laguerre-Polya Class
![Page 17: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/17.jpg)
Historical BackgroundCurrent Research
Non-linear operators.
Example
Construct a degree 5 polynomial p(x) with zeros x = −1.
p(x) = (x + 1)(x + 1)(x + 1)(x + 1)(x + 1)= 1 + 5x + 10x2 + 10x3 + 5x4 + x5
= a0 + a1x + a2x2 + a3x3 + a4x5 + a5x5 .
Replace ak with 3a2k − 4ak−1ak+1 + ak−2ak+2.
p(x) becomes 3 + 35x + 105x2 + 105x3 + 35x4 + 3x5....and the zeros remain real and negative.
Lukasz Grabarek The Laguerre-Polya Class
![Page 18: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/18.jpg)
Historical BackgroundCurrent Research
Non-linear operators.
Example
Construct a degree 5 polynomial p(x) with zeros x = −1.
p(x) = (x + 1)(x + 1)(x + 1)(x + 1)(x + 1)= 1 + 5x + 10x2 + 10x3 + 5x4 + x5
= a0 + a1x + a2x2 + a3x3 + a4x5 + a5x5 .
Replace ak with 3a2k − 4ak−1ak+1 + ak−2ak+2.
p(x) becomes 3 + 35x + 105x2 + 105x3 + 35x4 + 3x5....and the zeros remain real and negative.
Lukasz Grabarek The Laguerre-Polya Class
![Page 19: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/19.jpg)
Historical BackgroundCurrent Research
Non-linear operators.
Example
Construct a degree 5 polynomial p(x) with zeros x = −1.
p(x) = (x + 1)(x + 1)(x + 1)(x + 1)(x + 1)= 1 + 5x + 10x2 + 10x3 + 5x4 + x5
= a0 + a1x + a2x2 + a3x3 + a4x5 + a5x5 .
Replace ak with 3a2k − 4ak−1ak+1 + ak−2ak+2.
p(x) becomes 3 + 35x + 105x2 + 105x3 + 35x4 + 3x5.
...and the zeros remain real and negative.
Lukasz Grabarek The Laguerre-Polya Class
![Page 20: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/20.jpg)
Historical BackgroundCurrent Research
Non-linear operators.
Example
Construct a degree 5 polynomial p(x) with zeros x = −1.
p(x) = (x + 1)(x + 1)(x + 1)(x + 1)(x + 1)= 1 + 5x + 10x2 + 10x3 + 5x4 + x5
= a0 + a1x + a2x2 + a3x3 + a4x5 + a5x5 .
Replace ak with 3a2k − 4ak−1ak+1 + ak−2ak+2.
p(x) becomes 3 + 35x + 105x2 + 105x3 + 35x4 + 3x5....and the zeros remain real and negative.
Lukasz Grabarek The Laguerre-Polya Class
![Page 21: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/21.jpg)
Historical BackgroundCurrent Research
Non-linear operators.
Example (continued...)
p(x) = 1 + 5x + 10x2 + 10x3 + 5x4 + x5
= a0 + a1x + a2x2 + a3x3 + a4x5 + a5x5 .
The zeros of p(x) remain real and negative if ak is replacedwith:
a2k − ak−1ak+1,
p(x) becomes 1 + 15x + 50x2 + 50x3 + 15x4 + x5
3a2k − 4ak−1ak+1 + ak−2ak+2,
p(x) becomes 3 + 35x + 105x2 + 105x3 + 35x4 + 3x5
10a2k − 15ak−1ak+1 + 6ak−2ak+2 − ak−3ak+3,
p(x) becomes 10 + 100x + 280x2 + 280x3 + 100x4 + 10x5
and infinitely many others....
Lukasz Grabarek The Laguerre-Polya Class
![Page 22: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/22.jpg)
Historical BackgroundCurrent Research
Non-linear operators.
Example (continued...)
p(x) = 1 + 5x + 10x2 + 10x3 + 5x4 + x5
= a0 + a1x + a2x2 + a3x3 + a4x5 + a5x5 .
The zeros of p(x) remain real and negative if ak is replacedwith:
a2k − ak−1ak+1,
p(x) becomes 1 + 15x + 50x2 + 50x3 + 15x4 + x5
3a2k − 4ak−1ak+1 + ak−2ak+2,
p(x) becomes 3 + 35x + 105x2 + 105x3 + 35x4 + 3x5
10a2k − 15ak−1ak+1 + 6ak−2ak+2 − ak−3ak+3,
p(x) becomes 10 + 100x + 280x2 + 280x3 + 100x4 + 10x5
and infinitely many others....
Lukasz Grabarek The Laguerre-Polya Class
![Page 23: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/23.jpg)
Historical BackgroundCurrent Research
Non-linear operators.
Example (continued...)
p(x) = 1 + 5x + 10x2 + 10x3 + 5x4 + x5
= a0 + a1x + a2x2 + a3x3 + a4x5 + a5x5 .
The zeros of p(x) remain real and negative if ak is replacedwith:
a2k − ak−1ak+1,
p(x) becomes 1 + 15x + 50x2 + 50x3 + 15x4 + x5
3a2k − 4ak−1ak+1 + ak−2ak+2,
p(x) becomes 3 + 35x + 105x2 + 105x3 + 35x4 + 3x5
10a2k − 15ak−1ak+1 + 6ak−2ak+2 − ak−3ak+3,
p(x) becomes 10 + 100x + 280x2 + 280x3 + 100x4 + 10x5
and infinitely many others....
Lukasz Grabarek The Laguerre-Polya Class
![Page 24: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/24.jpg)
Historical BackgroundCurrent Research
Non-linear operators.
Example (continued...)
p(x) = 1 + 5x + 10x2 + 10x3 + 5x4 + x5
= a0 + a1x + a2x2 + a3x3 + a4x5 + a5x5 .
The zeros of p(x) remain real and negative if ak is replacedwith:
a2k − ak−1ak+1,
p(x) becomes 1 + 15x + 50x2 + 50x3 + 15x4 + x5
3a2k − 4ak−1ak+1 + ak−2ak+2,
p(x) becomes 3 + 35x + 105x2 + 105x3 + 35x4 + 3x5
10a2k − 15ak−1ak+1 + 6ak−2ak+2 − ak−3ak+3,p(x) becomes 10 + 100x + 280x2 + 280x3 + 100x4 + 10x5
and infinitely many others....
Lukasz Grabarek The Laguerre-Polya Class
![Page 25: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/25.jpg)
Historical BackgroundCurrent Research
Non-linear operators.
Example (continued...)
p(x) = 1 + 5x + 10x2 + 10x3 + 5x4 + x5
= a0 + a1x + a2x2 + a3x3 + a4x5 + a5x5 .
The zeros of p(x) remain real and negative if ak is replacedwith:
a2k − ak−1ak+1,
p(x) becomes 1 + 15x + 50x2 + 50x3 + 15x4 + x5
3a2k − 4ak−1ak+1 + ak−2ak+2,
p(x) becomes 3 + 35x + 105x2 + 105x3 + 35x4 + 3x5
10a2k − 15ak−1ak+1 + 6ak−2ak+2 − ak−3ak+3,p(x) becomes 10 + 100x + 280x2 + 280x3 + 100x4 + 10x5
and infinitely many others....
Lukasz Grabarek The Laguerre-Polya Class
![Page 26: The Laguerre-Polya Class´lgrabarek/LG/ARCS.pdfEdmond Laguerre and George Polya´ Paul Turan´ George Csordas, Richard Varga, Timothy Norfolk The Riemann Hypothesis (1859) The most](https://reader033.vdocument.in/reader033/viewer/2022053021/5f8b57fc7de61a4cae528a16/html5/thumbnails/26.jpg)
Historical BackgroundCurrent Research
Non-linear operators.
The Main Result
Theorem (Grabarek (2010))
Let ϕ(x) =∑ω
k=0 akxk ,0 ≤ ω ≤ ∞, be a function in theLaguerre-Polya class. If the zeros of ϕ(x) are real and negative,then the zeros remain real and negative after replacing ak with(
2p − 1p
)a2
k +
p∑j=1
(−1)j(
2pp − j
)ak−jak+j (p = 1,2,3, . . .) .
Lukasz Grabarek The Laguerre-Polya Class