Kigakree Then, one has Proof. Since there can be more than one -analogue of any classical expression, we can define another -analogue for the Elzaki eelzaki. Since then from 39we have Thus, from Definition 1Let denote the th -derivative of the function. Moreover, it is easily transforj that where, andare real numbers. Hence, the -transform of 76 is Further simplifications yield The inverse -transform in 61 gives the solution 4. On a -Analogue of the Elzaki Transform Called Mangontarum -Transform Ifthen by Theorem 3 Thus, taking the Mangontarum -transform of tdansform first kind of both sides of 68 gives us From the inverse -transform in 61we have the solution.
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Mikashicage View at Google Scholar T. The author declares that there is no conflict of interests regarding the publication of this paper. Theorem 19 -derivative of transforms. Some properties and interesting formulae are derived and presented. Now, by 28 and 27Similarly, Now, if we define the hyperbolic -sine and -cosine functions as we have the next theorem which presents the transforms of the -trigonometric ekzaki.
View at Google Scholar V. Ifthen the convolution identity for the -Laplace transform of the first kind in [ 10 ] is Now, from 47 and 50Thus, we have the following convolution identity for the Mangontarum -transform of the first kind. To obtain this, the following definition is essential. Ifwe say that is an inverse Mangontarum -transform of the first kindor an inverse -transform of the functionand we write.
For a positive integer andMuch is yet to be discovered regarding the Mangontarum -transforms. Moreover, by 24 and 29we have where. In this paper, we will define two kinds of -analogues of the Elzaki transform, and to differentiate them from other elzakj -analogues, we will refer to these transforms as Mangontarum -transforms. Fundamental properties of this transform were already established by Elzaki et al.
Theorem 4 duality relation. Ifwith -Laplace transforms of the first kind andrespectively, and Mangontarum -transforms of the first kind and transtorm, respectively, then From [ 10 ], the -derivative of the -Laplace transform of the first kind is Replacing with and applying Theorem 6 yield Hence, the following theorem is easily observed.
The author would like to thank the academic editor for his invaluable work during the editorial workflow and the referees for their corrections and suggestions which helped improve the clarity of this paper. Theorem 18 trannsform of -derivatives.
This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Researchers are encouraged to further investigate other applications of these -transforms, especially the second kind. Discrete Dynamics in Nature and Society Now, suppose 43 holds for.
The following statements are true: Taking the Mangontarum -transform of the first kind of both sides of this equation yields Let. Clearly, The expressions in 9 are called -integer-falling factorial of of order-factorial ofand -binomial coefficient or Gaussian polynomialrespectively. Theorem 16 transform of -trigonometric functions.
Other important tools in this sequel are the Jackson -derivative and the definite Jackson -integral Note that from 14 and 15we have Furthermore, given the improper -integral of we trqnsform see [ 10 ]. This motivates us to define a -analogue for the Elzaki transform in [ 1 ]. Abstract Two transdorm of the Elzaki transform, called Mangontarum -transforms, are introduced in this paper.
Subscribe to Table of Contents Transforrm. The -difference calculus or quantum calculus was first studied by Jackson [ 15 ], Carmichael [ 16 ], Mason [ 17 ], Adams [ 18 ], and Trjitzinsky [ 19 ] in the early 20th century. The study of the -analogues of classical identities is a popular topic among mathematicians and physicists. For instance, the two -analogues for the exponential function denoted by and are given, respectively, by with.
Since then from 39we have Thus, from Definition 1Let denote the th -derivative of the function. Also, note that there were no applications shown for the Mangontarum -transform of the second kind in eelzaki paper. Comtet, Advanced CombinatoricsD. Consider the first degree -differential equation: Then the Mangontarum -transform of the second kind satisfies the relation. Find the solution of the equation where and with and.
The book of Kac and Cheung [ 9 ] is a good source for further details of quantum calculus. To receive news and publication updates for Discrete Dynamics in Nature and Society, enter your email address in the box below. The Mangontarum -transform of the second kind, denoted byis defined by over the setwhere and. Observe that linearity also holds for the inverse -transform of the function. Ifthen by Theorem 3 Thus, taking the Mangontarum -transform of the first kind of both sides of 68 gives us From the inverse -transform in 61we have the solution.
Furthermore, the -binomial coefficient can be expressed as Note that the transition of any classical expression to its -analogue is not unique. Then, we have the following theorem. Reidel Publishing, Dordretcht, The Neatherlands, Application of In this section, we will consider applications of the Mangontarum -transform of the first kind to some -differential equations.
For example, for any integers and with andwe have the following -analogues of the integerfalling factorialfactorialand binomial coefficientrespectively: Let Then, we have the following definition. Then, by Theorem 17Hence, the following theorem is obtained. TOP Related.
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Mikashicage View at Google Scholar T. The author declares that there is no conflict of interests regarding the publication of this paper. Theorem 19 -derivative of transforms. Some properties and interesting formulae are derived and presented. Now, by 28 and 27Similarly, Now, if we define the hyperbolic -sine and -cosine functions as we have the next theorem which presents the transforms of the -trigonometric ekzaki. View at Google Scholar V.
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