Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 253 https://internationalpubls.com Jacobi Matrix Polynomial and its Integral Results V.Sri Lakshmi1, N. Srimannarayana2*, M.V. Chakradhar Rao3, M. Radha Madhavi4, D.K.Pavan Kumar5 1Research Scholar, Department of Engineering Mathematics, College of Engineering, Koneru Lakshmaiah Education Foundation, Vaddeswaram, Guntur (Dt.,), A.P., India. Associate Professor in the Department of Mathematics, G.Pulla Reddy Degree and P.G. College, Mehdipatnam, Hyderabad, Telangana, India. Email: kanurisrilakshmi@gmail.com 2Professor, Department of Engineering Mathematics, College of Engineering, Koneru Lakshmaiah Education Foundation, Vaddeswaram, Guntur (Dt.,), Andhra Pradesh, India-522502. Email: sriman72@kluniversity.in 3M.V.Chakradhara Rao, Professor, Department of Mathematics, Presidency University, Bangalore, India. Email: chakrimv@yahoo.com 4Professor, Department of Engineering Mathematics, College of Engineering, Koneru Lakshmaiah Education Foundation, Vaddeswaram, Guntur (Dt.,), Andhra Pradesh, India-522502 Email: mrmadhavi5@gmail.com 5Asst.Professor, Department of Mathematics, Seshadri Rao Gudlavalleru Engineering College, Gudlavalleru - 521356, Andhra Pradesh, India. E-Mail: krish.pav@gmail.com Article History: Received: 18-10-2024 Revised: 02-12-2024 Accepted: 10-12-2024 Abstract: Introduction: Recent advancements in matrix polynomial structures associated with special functions have gained significant traction, showcasing a diverse array of applications across various engineering disciplines. This paper primarily aims to explore and derive multiple integral representations for the modified Jacobi Matrix Polynomial. Specifically, we will delve into both finite and infinite single integral representations, as well as double integral representations of the polynomial. By examining these integral forms, we seek to enhance the understanding of their mathematical properties and implications, thereby providing valuable insights that can be leveraged in practical engineering scenarios. The exploration of these representations not only enriches the theoretical framework surrounding modified Jacobi Matrix Polynomials but also highlights their potential utility in solving complex problems within the engineering domain. Keywords : Finite and Infinite Single Integral Representation, Hypergeometric function, Jacobi Polynomial (Modified), Jacobi matrix polynomial 1. Introduction In the past few years, many authors extended classical polynomials to matrix polynomials. Many authors generalized the hypergeometric series, Appell’s hypergeometric functions also to matrix version of the functions. Recently, a good number of authors studied matrix version of Jacobi, Hermite, Legendre and other polynomials [1]-[4]. The theory of matrix polynomials provides a way to solve many problems in mathematical physics which have real time applications. L.Jodar [5]-[8] introduced matrix form of Laguerre and Hermite and Hypergeometric function. Subhi Khan and others mailto:kanurisrilakshmi@gmail.com mailto:sriman72@kluniversity.in mailto:chakrimv@yahoo.com mailto:mrmadhavi5@gmail.com mailto:krish.pav@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 254 https://internationalpubls.com extended Laguerre polynomials with two variables [9]. Parihar and Patel [10] introduced the modified Jacobi polynomials and derived generating function and recurrence relations of modified Jacobi polynomial. Later on, Sri Lakshmi, V [4] introduced matrix form of modified Jacobi polynomial and derived generating function, recurrence relations of matrix version of modified Jacobi polynomial. Srimanarayana, N et.al [11], [12], derived integral representations for Generalized Hypergeometric function and modified Konhauser’s polynomial. In the present study finite and infinite integral representations and double integral representations has been established for ( ) ( , )A nJ x w . Useful functions and Integral Formulae: In the present article, we abide by the rules of matrix theory. Assume 0 1, ,..., m m nP P P C  , where iP be the matrix of size ‘m’ and the matrix polynomial of degree ‘n’ as 1 1 1 0( ) ...n n n n nf z P z P z P z P− −= + + + + , where nP is not a null matrix. Also, I and O be the identity and null matrices in m mC  . For a matrix m mP C  , ( )P is the spectrum of P and the matrix P is a positive stable matrix. In [13], for all m mP C  and P n I+ is invertible, where ‘n’ is an integer, then the Pochhammer symbol is defined by 1 ( )...( ( 1) ); 0 ( ) ( ) ( ) ; 0 ; 0 n P P I P n I n P P nI P n I n − + + −   =  +    = (1.1) For preliminary matrix version of Gamma, Beta, and Hypergeometric functions one can refer [2], [5], [6], [7], and [13]. In [10], Parihar and Patel defined modified Jacobi Polynomial using a difference operator [14]. Later, Sri Lakshmi, V, et.al extended the same as modified Jacobi Matrix polynomial, ( ) ( , )A nJ x w as follows [4]. ( ) 2 1 ( ) ( , ) , ; ; ! A n n A I x J x w F nI I A w n w +   = − +    (1.2) ( ) 0 ( ) ( ) ! !( ) r n r n r r r x nI w A I w n r I A=   −   +   = +  (1.3) To obtain different integral representations, the following well-known results have been used: Maclaurin’s theorem is ( ) 0 (0) ( ) ! l l l f s f s l  = = (1.4) so that the coefficients ( ) (0); 0,1, 2,lf l = − − − are obtained by the integrals Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 255 https://internationalpubls.com (0 ) ( ) 1 ! ( ) (0) ; 0,1,2... 2 l n l f s ds f s i s + + = = (1.5) For Re( )m and Re( ) 0n m−  , we have 1 1 1 0 ( ) ( ) (1 ) ( ) ( ) ( ) j m j n m j m n d n m n m   + − − − = −   −  (1.6) 2 2 1( ) t nn e t dt  − − −  =  (1.7) If Real( , ) 0m n  , then 1 1 1 1 1 0 0 (1 ) (1 ) ( , ) (1 ) k k l k l x y y dxdy k l xy  − − + − − − = −   (1.8) If Real (k,l)>0, Real(𝜆)>0 then 1 1 1 ( ) ( ) ( ) (1 ) ( ) k l A k l u v u v dudv k l    − − −    − − =  + + (1.9) where ‘A’ is the area lies between u, v ≥ 0 and u+v ≤ 1. /2 2 1 2 1 0 ( , ) 2 (sin ) (cos )a ba b t t dt   − −=  (1.10) 2 2 1 ( ) 2 2 2 r r r r    +    =         (1.11) provided Re(s)>0 and Re (𝛼)>0 [15], [16] 2. INTEGRAL REPRESENTATION FOR ( ) ( , )A nJ x w Integral Representation by a Contour Integral By the generating function of ( ) ( , )A nJ x w , we have ( ) 1 1 0 ( , ) ( , ; ) ( ) A n tn n n J x w x t e F I A wt I A w  = = + − +  (2.1) Assume 1 1( ) ( , ; )u x f u e F I A wu w = + − (2.2) Using Maclaurin’s theorem (1.4) and using (1.5), we arrive at the following: Theorem: Assume ‘ A ’ be any square matrix of order ‘n’, 0A Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 256 https://internationalpubls.com 0 ( ) 1 1 1 ( ) ( , ) , ; 2 A n un n A I x J x w u e F I A wu du i w + − −+   = + −     Where the contour integral is around the u-plane in anti-clockwise sense. Proof: Using Maclaurin’s Theorem, we have ( ) 0 (0) ( ) ! n n n f u f u n  = = 0 ( ) 1 ! ( ) (0) ; 0,1, 2 n n n f u f du n i u + +  = = −−− If ( ) 1 1 0 ( , ) ; ; ( ) A u nn n n J x wx e F I A wu u w I A  =   + − =  +   0( ) 1 ! ( , ) ! ( ) ( ) 2 A n n n n J x w n f u du I A i u + +  = +  , which leads to 0 ( ) 1 1 1 ( ) ( , ) , ; 2 A n un n A I x J x w u e F I A wu du i w + − −+   = + −     2. Real Integral Representation Theorem: If ‘A’ be any square matrix of order n n and ( ) ( , )A nJ x w be the modified Jacobi matrix polynomial, then the real integral representations of this polynomial are as follows: ( ) , 0 0 ( ) ( ) ( , ) ( ) !( ) r A n r n r m r x w A I w J x w cis m r n d r I A      =   −  +   = + − +   (3.1) Proof: Using (1.14), it reduces to (by choosing the contour iu e = ( )2 ( ) ( 1) 00 ( ) ( , ) ! (2 ) !( ) i r r i A e i n in r n r r x w e A I w J x w e e ie d n i r I A         − − =     +   = +  2 0 00 ( ) ( ) ( ) ! (2 ) ! ! ( ) r ri i m nin r m r r x w e A I e w e d n m r I A         − = =   −  +   = +   By changing the order of summation and integration, we obtain Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 257 https://internationalpubls.com 2 ( ) , 0 0 ( ) ( ) ! (2 ) ! ( ) r m n r in r r m r x w A I w e d n r I A      − + =   −  +   = +   2 , 0 0 ( ) ( ) ( ) (2 ) ! ( ) r n r r m r x w A I w cis m r n d r I A     =   −  +   = + − +   Consequently, we arrive at ( ) , 0 0 ( ) ( ) ( , ) ( ) !( ) r A n r n r m r x w A I w J x w cis m r n d r I A      =   −  +   = + − +   4. Finite Single Integral Representation Theorem 1: Assume ‘ A ’ be any square matrix of order ‘n’, 0A . If Re( )a and Re( )b a− are positive, then ( ) ( ) ( ) 1 1 1 ( ) 3 2 0 , , ;(1 ) ( , ) , ; a b a A n x b nI bt t J x w F wt dtw a b a I A a − − −    −−  =    − +   (4.1) Proof: By using (1.3) ( ) ( ) 0 ( ) ( , ) ! ! ( ) r n r A n r n r r x nI w A I w J x w n r I A=   −   +   = +  ( ) ( ) ( ) ( ) ( )0 ( ) ! !( ) r n r r r n r r r r r x nI w b a A I w n r I A a b=   −   +   = +  On using (1.5), we have ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 0 0 ( ) (1 ) ! !( ) r n r r a r b an r r r r x nI w b b A I w t t dt n r I A a a b a + − − − =   −   +   = − +   −   By interchanging the order of integration and summation, we have ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 00 ( ) ( ) (1 ) ! ! ( ) r n r r a b a n r r r r x nI b wt b A I w t t dt a b a n r I A a − − − =   −   +   = −   − +  Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 258 https://internationalpubls.com ( ) ( ) ( ) 1 1 1 3 2 0 , , ; (1 ) ,( ); a b a x b nI b t t F wt dtw a b a a I A − − −    − = −    − +   Theorem 2: Assume ‘ A ’ be any square matrix of order ‘n’, 0A If real part of 1 , 2 a b− , then /2 ( ) 2 1 2 1 2 2 4 3 0 1 , , , ;2 ( ) ( ) ( , ) (sin ) (cos ) sin cos2 2 ! ( ) ( ) ( ), , ; A a b n x a b a b nIa b I A J x w t t F w t t dtw n a b I A a b  − − + + +  − + +  =    +   Proof: By the equation (1.3) of ( ) ( , )A nJ x w , we have 2 1 ( ) ( , ) , ; ; ! A n n I A x J x w F nI I A w n w +   = − +    ( ) 0 ( ) ( 2 ) ( ) ( ) ! ( ) ! ( ) ( ) ( 2 ) r n r n r rr r r r r x nI w I A a b r a bw n I A r a b a b r=   −   +  + +  = +  + +  ( ) /2 2 2 1 2 2 1 0 0 ( ) 2 ( 2 ) (sin ) (cos ) ! ( ) ! ( ) ( ) r n r a r b rn r r r x nI w I A a b rw t t dt n I A r a b  + − + − =   −   +  + +  = +     On interchanging the order of integration and summation, we have ( )/2 2 1 2 1 2 2 00 1 ( ) ( ) 2 2 2 (sin ) (cos ) (sin ) (cos ) ! ( ) ( ) ( ) ! ( ) ( ) n r a b r r rn r r r r r r r x a b a b nI I A a b w t t t t w dt n a b I A r a b  − − = + + +      −       +  +       =   +  /2 2 1 2 1 2 2 4 3 0 1 , , , ;2 ( ) ( ) (sin ) (cos ) sin cos2 2 ! ( ) ( ) ( ), , ; a b x a b a b nIa b I A t t F w t t dtw n a b I A a b  − − + + +  − + +  =    +   Hence the proof. 5. Infinite Single Integral Representation If ‘ A ’ be any square matrix of order n n and ( ) ( , )A nJ x w be the modified Jacobi matrix polynomial, then infinite single integral representation of this polynomial is as follows: Theorem Assume A be any square matrix of order ‘n’ 0A . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 259 https://internationalpubls.com 2( ) 2 1 2 2 2 , ;( ) ( , ) ( ) ! , ; A t an n x nII A J x w e t F wt dtw a n a I A  − − −   −+  =   +   (5.1) Proof: From the equation (1.3) of ( ) ( , )A nJ x w , we have 2 1 ( ) ( , ) , ; ; ! A n n I A x J x w F nI I A w n w +   = − +    ( ) 0 ( ) ( ) ! (a) ( ) ! ( ) r n r n r r r r x nI w a r I A w n I A r a=   −  +  +   =  +  ( ) 2 2 2 1 0 ( ) ! ( ) ( ) ! ( ) r n r t a rn r r r r x nI w I A w e t dt n a I A r a  − + − = −   −   +   =  +   On interchanging the order of integration and summation, we have ( ) 2 2 2 1 0 ( ) ( ) ( ) ! ( ) ( ) ! r n r t an r r r r x nI wt I A w e t dt a n a I A r  − − =−   −   +   =  +  2 2 1 2 2 2 , ;( ) ( ) ! , ; t an x nII A e t F wt dtw a n a I A  − − −   −+  =   +   Hence the theorem. 6. Double integral representation Theorem: Assume ‘ A ’ be any square matrix of order ‘n’, 0A . If real part of a, b, 𝜆 >0 then ( ) 1 1 1 4 3 1 , , , ;( )( ) ( , ) (1 ) 2 2 2 ! ( ) ( ) ( ) , , ; A a b a bn n A x nII A J x w u v u v F uvw dudvw n a b a b a b I A      − − − − −   − + +  = − −     − − +   (6.1) Proof: From the equation (1.3) of ( ) ( , )A nJ x w we have ( ) 2 1 ( ) ( , ) , ; ; ! A n n I A x J x w F nI I A w n w +   = − +    ( ) 1 1 1 0 1 ( )( ) 2 2 2 (1 ) ! ( ) ( ) ( ) ( ) ! ( ) ( ) r n r a r b r a bn r r r r r r r A x nI w I A w u v u v du dv n a b a b I A r a b      + − + − − − − =       − +       +       = − −    − − +   Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 260 https://internationalpubls.com On interchanging the order of integration and summation, we have 1 1 1 4 3 1 , , , ;( )( ) (1 ) 2 2 2 ! ( ) ( ) ( ) , , ; a b a bn A x nII A u v u v F uvw dudvw n a b a b a b I A      − − − − −   − + +  = − −     − − +   Hence the theorem. Special Cases: If ‘ A ’ be any square matrix of order n n and ( ) ( , )A nJ x w be the modified Jacobi matrix polynomial, then the applications of the above theorems of this polynomial are as follows: Theorem: 1 1 ( ) ( ) 0 ( , ) ( , ) (1 ) a A A n n t J x w J x wt dt t − = − Proof: By setting a = b, in (1.15), we come across at 1 1 ( ) 00 ( ) ( ) ( ) ( , ) (1 ) ! ! ( ) r ra n A n r n r r x nI wt t A I w J x w dt t n r I A − =   −   +   = − +  1 1 ( ) 0 ( , ) (1 ) a A n t J x wt dt t − = − Theorem: 1 1 ( ) ( ) 0 ( , ) ( ) (1 ) a A A n n t J x w L xt dt t − = − Proof: By considering 0w→ and setting a b= in (1.15), we arrive at the following: 1 1 ( ) 00 ( ) ( ) ( ) ( , ) (1 ) ! ! ( ) a rn A n r n r r t A I nI xt J x w dt t n n I A − = + − = − +  1 1 ( ) 0 ( ) (1 ) a A n t L xt dt t − = − Where ( ) ( )A nL x is the Laguerre matrix polynomial of one variable [2]. Theorem: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 261 https://internationalpubls.com 2( ) 2 1 2 1 2 ;( ) ( , ) , ;! ( ) A t an n nII A J x w e t F xt dt a I An a  − − − − + =   +    Proof: By assuming 0w→ , we have 2 2 ( ) 2 1 0 ( ) ( ) ( ) ( , ) ! ( ) ( ) ( ) ! rn A t an r n r r r I A nI xt J x w e t dt n a a I A r  − − =− + − =  +  2 2 1 2 1 2 ;( ) , ;! ( ) t an nII A e t F xt dt a I An a  − − − − + =   +    CONCLUSION Recent advancements in matrix polynomial structures associated with special functions have gained significant traction, showcasing a diverse array of applications across various engineering disciplines. This paper primarily intended to explore and derive multiple integral representations for the modified Jacobi Matrix Polynomial. Specifically, we will delve into both finite and infinite single integral representations, as well as double integral representations of the polynomial. By examining these integral forms, we seek to enhance the understanding of their mathematical properties and implications, thereby providing valuable insights that can be leveraged in practical engineering scenarios like applying Laplace Transforms. 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