• Title/Summary/Keyword: mean value theorems

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ON THE r-TH HYPER-KLOOSTERMAN SUMS AND ITS HYBRID MEAN VALUE

  • Zhang, Tianping;Zhang, Wenpeng
    • Journal of the Korean Mathematical Society
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    • v.43 no.6
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    • pp.1199-1217
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    • 2006
  • The main purpose of this paper is using the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet L-functions to study the hybrid mean value of the T-th hyper-Kloosterman sums Kl(h, k+1, r;q) and the hyper Cochrane sums C(h, q; m, k), and give an interesting mean value formula.

MEAN VALUES OF THE HOMOGENEOUS DEDEKIND SUMS

  • WANG, XIAOYING;YUE, XIAXIA
    • Korean Journal of Mathematics
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    • v.23 no.4
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    • pp.571-590
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    • 2015
  • Let a, b, q be integers with q > 0. The homogeneous Dedekind sum is dened by $$\Large S(a,b,q)={\sum_{r=1}^{q}}\(\({\frac{ar}{q}}\)\)\(\({\frac{br}{q}}\)\)$$, where $$\Large ((x))=\{x-[x]-{\frac{1}{2}},\text{ if x is not an integer},\\0,\hspace{75}\text{ if x is an integer.}$$ In this paper we study the mean value of S(a, b, q) by using mean value theorems of Dirichlet L-functions, and give some asymptotic formula.

ON THE γ-TH HYPER-KLOOSTERMAN SUMS AND A PROBLEM OF D. H. LEHMER

  • Tianping, Zhang;Xifeng, Xue
    • Journal of the Korean Mathematical Society
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    • v.46 no.4
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    • pp.733-746
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    • 2009
  • For any integer k $\geq$ 2, let P(c, k + 1;q) be the number of all k+1-tuples with positive integer coordinates ($a_1,a_2,...,a_{k+1}$) such that $1{\leq}a_i{\leq}q$, ($a_i,q$) = 1, $a_1a_2...a_{k+1}{\equiv}$ c (mod q) and 2 $\nmid$ ($a_1+a_2+...+a_{k+1}$), and E(c, k+1; q) = P(c, k+1;q) - $\frac{{\phi}^k(q)}{2}$. The main purpose of this paper is using the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet L-functions to study the hybrid mean value of the r-th hyper-Kloosterman sums Kl(h,k+1,r;q) and E(c,k+1;q), and give an interesting mean value formula.

ON OPIAL-TYPE INEQUALITIES VIA A NEW GENERALIZED INTEGRAL OPERATOR

  • Farid, Ghulam;Mehboob, Yasir
    • Korean Journal of Mathematics
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    • v.29 no.2
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    • pp.227-237
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    • 2021
  • Opial inequality and its consequences are useful in establishing existence and uniqueness of solutions of initial and boundary value problems for differential and difference equations. In this paper we analyze Opial-type inequalities for convex functions. We have studied different versions of these inequalities for a generalized integral operator. Further difference of Opial-type inequalities are utilized to obtain generalized mean value theorems, which further produce various interesting derivations for fractional and conformable integral operators.

A STUDY OF THE RIGHT LOCAL GENERAL TRUNCATED M-FRACTIONAL DERIVATIVE

  • Chauhan, Rajendrakumar B.;Chudasama, Meera H.
    • Communications of the Korean Mathematical Society
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    • v.37 no.2
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    • pp.503-520
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    • 2022
  • We introduce a new type of fractional derivative, which we call as the right local general truncated M-fractional derivative for α-differentiable functions that generalizes the fractional derivative type introduced by Anastassiou. This newly defined operator generalizes the standard properties and results of the integer order calculus viz. the Rolle's theorem, the mean value theorem and its extension, inverse property, the fundamental theorem of calculus and the theorem of integration by parts. Then we represent a relation of the newly defined fractional derivative with known fractional derivative and in context with this derivative a physical problem, Kirchoff's voltage law, is generalized. Also, the importance of this newly defined operator with respect to the flexibility in the parametric values is described via the comparison of the solutions in the graphs using MATLAB software.