View of A STUDY ON P-TRANSFORM OF MULTIVARIABLE POLYNOMIALS

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ACCENT JOURNAL OF ECONOMICS ECOLOGY & ENGINEERING Peer Reviewed and Refereed Journal IMPACT FACTOR: 2.104 (ISSN NO. 2456-1037) Vol.03, Issue 09, Conference (IC-RASEM) Special Issue 01, September 2018 Available Online: www.ajeee.co.in/index.php/AJEEE

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A STUDY ON P-TRANSFORM OF MULTIVARIABLE POLYNOMIALS Mrs. Namrata N. Nadgauda

Assistant Prof., Department of Mathematics, B.E. Society’s Lt. K.G. Kataria College, Daund Maharastra India

Abstract In the present paper, the author istoderive the P-transform of Multivariable H- function and general polynomials.P-transform is useful in reaction theory in astrophysics.

P-transform is generalization of many integral transforms, Multivariable H-function and general Multivariable polynomials are general in nature. These results discussed here can be used to investigate wide class of new and known results, hitherto scattered in the literature. For the sake of illustration, some special case have also been mentioned here of our finding.

Keywords: P-transform, general class of multivariable polynomials, Multivariable H- function, Hermitee polynomial, Laguerre polynomial.

I. INTRODUCTION

The P-transform is defined and

represented in Kumar and Kilbas [1] as (1.1) where represents the kernel- function

(1.2) with

given by (1.2), P-transform is called type-1 is Ptransform. If we use

(1.3) For𝜏 ∈ ℂ, 𝜇 > 0, 𝜌 > 𝑅, 𝑎 > 0, 𝛽 > 1, On (1.

1) At that point we acquire a type-2 P- transform. Those P-transform about both sorts are characterized in the space 𝐿𝜏,𝑟 (0, ∞) comprising of the Lebesgue measurable unpredictable esteemed

works 𝑓 for which.

(1.4)

for 1 ≤ 𝑟 < ∞, 𝜏 ∈ 𝑅. The P-transform of both the types is obtained by using the pathway model of Mathai [5], Mathai and Haubold [6]. If 𝜇 = 1, 𝑎 = 1, 𝛽 → 1, we have

(1.5)

where is the kernel function of theKrätzel transform, introduced by Krätzel [12] and given by (1.5)

(1.6)

where

(1.7)

The transform in (1.6) and its several modifications were considered by many authors. Glaeske et al. [14]

considered a generalized version of the Krätzel transform and its compositions with fractional calculus operators on the spaces of Bonilla et al. [7, 8]

studies the Krätzel transform in the spaces Kilbas et al. [2]

obtained the asymptotic representation for the modified Kratzel function. Klibas et al. [3] studied the Krätzel function in (1.7) for all values of 𝜌 and established it in the terms of Fox’s H-function, when 𝜇 = 1, 𝜌 = 1, 𝑎 = 1, 𝑎𝑛𝑑 𝛽 → 1P-transform of both types reduces to the Meijer transform. For 𝜇 = 1, 𝜌 = 1, 𝑎 = 1, 𝑎𝑛𝑑 𝛽 → 1, along with 𝑥

replaced by in (1.2) and (1.3), we get (1.8)

where𝐾−𝜏 𝑡 is modified Bessel function of the third kind or the Mc-Donald function (see [4], sect. 7.2.2). Kilbas and Kumar [1]

considered (1.3) for 𝜇 = 1 and established its composition with fractional operators and represented it in terms of various generalized special functions.

The H-function of several complex variables, defined H. M. Srivastava and R.

Panda [10], we will define and represent it in the following from [1, page 252, equation (C.1)]

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2 where

Here,

stands for the sequence of 𝑟 ordered pairs

The general multivariable polynomials introduced and defined by Srivastava [15]

in the following form:

where are non-zero

arbitrary positive integer. The coefficients being arbitrary (real or complex) constants.

II. MAIN RESULTS Theorem

, be such 𝜌 > 0 in the case of a type 1P- transform then

Where and the coefficients are arbitrary constants, real or complex.

Theorem 2

be such 𝜌 ∈ 𝑅, 𝜌 ≠ 0 in the case of a type- 2 P-transform then

Where and the coefficients are arbitrary constants, real or complex.

Proof: To prove (2.1), we consider type-1 P-transform using (1.1) express H- function multivariable and general multivariablepolynomial with the help of (1.9), (1.12), we obtain

Changing the order of integrations and

series, taking and using

gamma function, we get

Now, solving inner integral with the help of beta function formula and then reinterpreting the result MellinBarnes contour integral in terms of h-function multivariable, we get the desired result.

On the similar lines, the proof of result (2.2) can be established using the

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3 definition of multivariable H-function (1.9) and type-2 P-transform (1.1).

III. SPECIAL CASES

On taking 𝑎 = 1, 𝜇 = 1 and 𝛽 → 1 in result (2.1), we have

The general multivariable polynomial decrease to general population for polynomial and multivariable H-function on Fox’s H-function, after that applying u=2 for (2. 1), (2. 2) and (3. 1), all population about polynomial lessens will Hermite polynomials [13] by setting.

Results in (ii), (iii) and (iv) are valid under the same conditions as are required for (2.1), (2.2) and (3.1) respectively.

Putting u=1 in (2.1), (2.2) and (3.1), general class of polynomial reduces to Laguerrre polynomials [13] by setting

Brings about (v), (vi) What's more (vii) need aid substantial under the same states as are needed for (2. 1), (2. 2) Also (3. 1) separately. The effects determined in this paper might without a moment's delay yield a vast number from claiming results, directing, including an extensive assortment for polynomials Furthermore Different uncommon capacities.

IV. CONCLUSION

In the display paper, we need gotten the effects have been produced As far as those result about multivariable H- function and An general class from claiming polynomials clinched alongside a conservative Furthermore exquisite with those assistance about Ptransform.

Practically of the effects got need been place over An conservative form, avoiding the event from claiming issues for math Furthermore Along these lines making them advantageous in provisions.

ACKNOWLEDGMENT

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4 The creators might Exceptionally appreciative will deserving referees to as much important suggestions which aided on attain spread change of the paper.

REFERCNCES

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