partial fraction expansion
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2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Jay Mehta ◽  
P. -Y Zhu

In this article, we shall prove a result which enables us to transfer from finite to infinite Euler products. As an example, we give two new proofs of the infinite product for the sine function depending on certain decompositions. We shall then prove some equivalent expressions for the functional equation, i.e. the partial fraction expansion and the integral expression involving the generating function for Bernoulli numbers. The equivalence of the infinite product for the sine functions and the partial fraction expansion for the hyperbolic cotangent function leads to a new proof of the functional equation for the Riemann zeta function.


2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Minghui You

AbstractBy introducing a kernel involving an exponent function with multiple parameters, we establish a new Hilbert-type inequality and its equivalent Hardy form. We also prove that the constant factors of the obtained inequalities are the best possible. Furthermore, by introducing the Bernoulli number, Euler number, and the partial fraction expansion of cotangent function and cosecant function, we get some special and interesting cases of the newly obtained inequality.


2019 ◽  
Vol 11 (3) ◽  
pp. 225-241
Author(s):  
Yih T. Tsay ◽  
Leang S. Shieh ◽  
R.E. Yates ◽  
S. Barnett

2019 ◽  
Vol 47 (4) ◽  
pp. 513-531 ◽  
Author(s):  
Panagiotis Bertsias ◽  
Costas Psychalinos ◽  
Brent J. Maundy ◽  
Ahmed S. Elwakil ◽  
Ahmed G. Radwan

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