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International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056

Volume: 12 Issue:10 | Oct 2025 www.irjet.net p-ISSN: 2395-0072

A Study on Integral Transform of the Generalized Lommel Wright KFunction

Sapna Chanderiya1, Dr. Indu bala bapna2

1Department of Mathematics, Maharshi Dayanand Saraswati University, Ajmer, Rajasthan, India

2 Department of Mathematics, Maharshi Dayanand Saraswati University, Ajmer, Rajasthan, India

Corresponding Author: sapnachanderiya5@gmail.com ***

Abstract – In the present work we introduces an integral transform of the lommel wright k- function. These transform are expressed in terms of the wright hypergeometric k-function. various interesting transform as the consequence of this method are obtained. which plays an important role to solve the differential equation and also some relations and results related to this generalized lommel wright k-function

Key Words: Generalized Lommel Wright K-function, Euler Beta Transform, Laplace Transform, K-Transform, Hankel Transform. MSC2020 classification:33E20,33B15,44A10,44A05,44A20

1.INTRODUCTION

Inrecentyearsthefractionalcalculushasbecomeoneofthemostrapidlygrowingresearchsubjectofmathematicalanalysis duetoitsnumerousapplicationsinvariouspartsofscienceaswellasmathematics.

ThetransformdefinedbythefollowingIntegralequation

iscalledtheEulerBetatransformwithpasacomplexparameter.

TheLaplacetransformofafunctionf(x)isdefinedby

Wherepisacomplexparameter.

Thetransform

iscalledtheK-transformwithpasacomplexparameterand iscalledthemodifiedBesselfunctionofthethirdkindorthe Macdonaldfunction,see(Mathaietal,2010,p.53).

TheHankeltransformofafunctionf(x)denotedbyg(p,v)definedas

where iscalledtheBessel-MaitlandfunctionorthemaitlandBesselfunction(Mathaietal,2010,p.22andp.56).

Nextfortheresultofthispaper.wetakeTheWrightHypergeometricfunctioninseriesform[9],denotedby isdefine as: wherethecofficient and arepositiverealnumberssuchthat.

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056

Volume: 12 Issue:10 | Oct 2024 www.irjet.net p-ISSN: 2395-0072

Gehlotandprajapatidefinedthelommelwrightkfunctionasfollows[5].

For

WetaketheWrightHypergeometrick-functioninseriesform[3],denotedby isdefineas:

Wherethecofficients and arepositiverealnumbersuchthat andslightlygeneralizedformis

Where isthegeneralisedhypergeometrick-functiondefinedby

Theseriesrepresentationofthegeneralizedlommelwrightk-functionisdefinedby

Where and isthek-gammafunctionintroducedbyDiaz andPariguan[6] givenby

withk-pochhammersymbol =

TheclassicalEulergammafunctionandgammak-functionarerelatedwithfollowingrelation

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056

Volume: 12 Issue:10 | Oct 2025 www.irjet.net p-ISSN: 2395-0072

fork=1generalizedlommelwrightk-functionreduceingeneralizedlommelwrightfunction.Form=1reducingingeneralized besselmaitlandk-function.form=k=1reducinginbesselmaitalandfunction.form=v=k=1and reducinginclaasicalbessel function.

Theorem 2.1.(Euler Beta Transform)

Let thentheEulerbetatransformofthegeneralizedlommelwrightkfunctionis

Proof-Onusing(8)intheintegrandof(10),weget

Nowusing(1)intheaboveequation,weget

Theorem 2.2.(Laplace Transform)

Let thenthelaplacetransformofthegeneralizedlommelwrightk-function is

Proof: Onusing(8)intheintegrandof(14),weget

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056

Volume: 12 Issue:10 | Oct 2025 www.irjet.net

Nowusing(2)intheaboveequationweget

Theorem 2.3. (K-Transform)

Let thenthek-transformofthegeneralizedlommelwrightk-functionis

Proof:Onusing(8)intheintegrandof(18),weget

Nowusing(3)intheaboveequationweget

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056

Volume: 12 Issue:10 | Oct 2025 www.irjet.net p-ISSN: 2395-0072

Theorem 2.4.(Hankel Transform)

Let thenthehankeltransformofthegeneralizedlommelwrightk-function is

Proof:Onusing(8)intheintegrandof(22),weget

Nowusing(4)intheaboveequationweget

Special Cases:

Inthissection,wegetsomeintegralformulasinvolvinglommelwrightk-functionasfollows.

Usingm=1,theresults(2.1)to(2.4)takeoftheform

Corollary 1.

Let Thenby(10)theEulerBetaTransform,weobtain:

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056

Volume: 12 Issue:10 | Oct 2025 www.irjet.net p-ISSN: 2395-0072

Corollary 2.

Let Thenby(14)theLaplaceTransform,weobtain:

Corollary 3.

Let Thenby(18)theK-Transform,weobtain:

Corollary 4.

Let Thenby(22)theHankelTransform,weobtain:

Lettingk=1,wehavethegeneralizedLommel-Wrightfunctionandthecorrespondingformulasarepresentedinsubsequent corollaries.

Corollary 5.

Let Thenby(10)theEulerBetaTransform,weobtain:

Corollary 6.

Let Thenby(14)theLaplaceTransform,weobtain:

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056

Volume: 12 Issue:10 | Oct 2025 www.irjet.net p-ISSN: 2395-0072

Corollary 7.

Let Thenby(18)theK-Transform,weobtain:

Corollary 8.

Let Thenby(22)theHankelTransform,weobtain:

Form=k=ψ=1and ℏ =0,thecorrespondingcorollariesareasgivenbelow Corollary 9.

Let Thenby(10)theEulerBetaTransform,weobtain:

Corollary 10.

Let Thenby(14)theLaplaceTransform,weobtain:

Corollary 11.

Let Thenby(18)theK-Transform,weobtain:

Corollary 12.

Let Thenby(22)theHankelTransform,weobtain:

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056

Volume: 12 Issue:10 | Oct 2025 www.irjet.net p-ISSN: 2395-0072

CONCLUSION: TheEulerBetaTransform,LaplaceTransform,K-Transform,HankelTransformforGeneralizedLommelwright k-functionwerederived.Whenwetakek=1wealsodeducetheresultsforLommelwrightfunction.

REFERENCES:

[1] MubeenS.andHabibullahG.M.,K-fractionalintegralsandapplications,Int.J.Compt.Math.Science,7(2)2012,pp.-89-94.

[2] J.Edward,ATreatiseontheintegralcalculus,Vol.2,ChelseaPublishingCompany,NewYork(1962).

[3] J.Pkonovska,TheoremsontheconvergenceofseriesingeneralizedLommel -Wrightfunctions,FractionalCalculusand AppliedAnalysis10,59-74(2007).

[4] K.B.Kachhia,J.C. Prajapti,On generalized fractional kinetic equation involving generalized Lommel-Wright functions,AlexandriaEngeneeringJournal,55(2016)29532957.

[5] K.S. Gehlot,J.C. Prajapati,Fractional calculus of generalized k-wright function,Journal of fractional calculus and applications,4(2013)283-289.

[6] R.Diaz,E.Pariguan,OnHypergeometricfunctionsandpochhammerk-symbol,DivulgacionesMathematics,15(2017),179192.

[7] A.M.Mathai,R.K.Saxena,H.J.Haubold,TheH-function:Theoryandapplications,NewYork,USA:Springer,2010.

[8] S.Mubeen,G.M.Habibullah,k-fractionalintegralsandapplication,Int.J.Contemp.Math.Sci.,7(2012),89–94.

[9] K.S.Gehlot,J.C.Prajapati,Fractionalcalculusofgeneralizedk-Wrightfunction,J.Fract.Calc.Appl.,4(2013),283–289.

[10] R.Diaz,E.Pariguan,OnhypergeometricfunctionsandPochhammerk-symbol,DivulgacionesMatematicas´,15(2007), 179–192.

[11] G.N.Watson,AtreatiseonthetheoryofBesselfunctions,2Eds.,London,NewYork:CambridgeUniversityPress,1944.

[12] D.L.Suthar,Compositionformulaeforthek-fractionalcalculusoperatorsassociatedwithk-WrightFunction,J.Math.,2020, 5471715.

[13] R.Diaz,C.Teruel,q,k-Generalizedgammaandbetafunctions,J.NonlinearMath.Phys.,12(2005),118–134.

[14] Choi, J. & Agarwal, P. 2013. Certain unified integrals associated with Bessel functions. Boundary Value Problems, art.number:95.

[15] Choi,J.,Mathur,S.&Purohit,S.D.2014.CertainnewintegralformulasinvolvingthegeneralizedBesselfunctions.Bulletin oftheKoreanMathematicalSociety,51(4),pp.995-1003.

[16] Kachhia,K.B.&Prajapati,J.C.2016.OngeneralizedfractionalkineticequationsinvolvinggeneralizedLommelWright functions.AlexandriaEngineeringJournal,55(3),pp.2953-2957.

[17] -

[18] Mathai, A.M. & Saxena, R.K. 1973. Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences.BerlinHeidelberg:Springer-Verlag.

[19] Mondal,S.R.&Nisar,K.S.2017.Certainunifiedintegralformulasinvolvingthegeneralizedmodifiedk-Besselfunctionof firstkind.CommunicationsoftheKoreanMathematicalSociety,32(1),pp.47-53.

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056

Volume: 12 Issue:10 | Oct 2025 www.irjet.net p-ISSN: 2395-0072

[20] Paneva- -Wrightfunctions.Fractional CalculusandAppliedAnalysis,10(1),pp.59-74.

[21] Rainville,E.D.1960.SpecialFunctions.NewYork:MacmillanCompany.

[22] Srivastava,H.M.&Manocha,H.L.1984.Atreatiseongeneratingfunctions.Chichester,WestSussex,England:E.Horwood& NewYork:HalstedPress.