Generalized multiple integral representations for a large family of polynomials with applications.pdf

applied mathematics and computation 261 (2015) 3947 contents lists available at sciencedirect applied mathematics and computation journal homepage: generalized multiple integral representations for a large family of polynomials with applications sebastien gaboury, richard tremblay dpartement dinformatique et mathmatique, universit du qubec chicoutimi, chicoutimi, qubec g7h 2b1, canada a r t i c l ei n f o msc: 33c45 33c65 keywords: hypergeometric polynomials linearization relations integral representations jacobi polynomials konhauser polynomials generalized sister celine s polynomials gamma function eulerian beta integral a b s t r a c t this paper aims to provide a natural generalization and unifi cation of a series of multiple in- tegral representations for special classes of hypergeometric polynomials recently obtained by several authors. this generalization is obtained by considering a very large family of hyperge- ometric polynomials. the multiple integral representations given in this paper may be viewed as linearization relationship for the product of two different members of the associated family of hypergeometric polynomials. 2015 elsevier inc. all rights reserved. 1. introduction almost four decades ago, srivastava 1 introduced and investigated the following general family of polynomials: sn n(z):= n n ? k=0 (n)nk k! an,kzk(n n0:= n 0;n n),(1.1) wherean,k n,k=0 is a suitably bounded double sequence of real or complex numbers, a denotes the greatest integer a r, and () denotes the pochhammer symbol defi ned by ():= ?(+) ?() = ? (+ 1).(+ n 1) (= n n; c), 1(= 0; c 0), n and c being, as usual, the set of positive integers and the set of complex numbers, respectively. moreover, it is understood conventionally that (0)0? 1. this last family of polynomials and their variants as sn n,m(z):= n n ? k=0 (n)nk k! am+n,kzk(m,n n0:= n 0;n n)(1.2) corresponding author. tel.: +1 418 545 5011. e-mail addresses: s1gabouruqac.ca (s. gaboury), rtremblauqac.ca (r. tremblay). /10.1016/j.amc.2015.03.088 0096-3003/ 2015 elsevier inc. all rights reserved. 40s. gaboury, r. tremblay/applied mathematics and computation 261 (2015) 3947 were widely studied by gonzalez et al. 2 and recently by lin et al. 3 (see also lin et al. 4,5). it is easy to see that sn n,0(z)= s n n(z). recently, altin et al. 6 investigated the following family of bivariate polynomials: sm,n n (x,y):= n n ? k=0 am+n,k xnnk (n nk)! yk k! (m,n n0:= n 0;n n),(1.3) which family contains as special cases the lagrangehermite polynomials, the lagrange polynomials and the hermitekamp de friet polynomials. the reader should read 611 for further details. it is easily observed by comparing defi nitions (1.1) and (1.2) that sm,n n (x,y)= xn n!s n n,m ? y (x)n ? (1.4) and sn n,m(z) ? ? ? ? y (x)n = n! xn sm,n n (x,y).(1.5) this shows the fact that the two-variable polynomials sm,n n (x,y)are essentially the same as the one-variable polynomials sn n,m(z). in the present investigation, we consider the following general family of polynomials: ?n;(k r);(lp) n;(ts);(mq) ?z; (r),(r),(p);(s),(s),(q)?=?n;k 1,.,kr;l1,.,lp n;t1,.,ts;m1,.,mq ? z ? ? ? ? 1,.,r;1,.,r;1,.,p 1,.,s;1,.,s;1,.,q ? (1.6) := n n ? j=0 (n)nj(1+1n)k1j.(r+rn)krj(1)l1j.(p)lpj (1+1n)t1j.(s+sn)tsj(1)m1j.(q)mqj zj j! (1.7) ?(k r),(lp),(ts),(mq) n 0;n n;(r),(r),(p);(s),(s),(q) c ? where (and throughout this paper) (kr) stands for the array of r parameters k1,.,kr, with similar interpretations for (lp), (ts), (mq), (r), (r), (p); (s), (s) and (q). this general family of polynomials contains, as special cases, several other families of polynomials. for example, the following relationships hold between the polynomials?n;(kr);(lp) n;(ts);(mq) ?z;( r),(r),(p);(s),(s),(q) ? and some simpler class of polynomials studied in 3,4,1214: rn,l1,l2 n,m1,m2(z;1,2,1,2):= n n ? j=0 (n)nj(1+ n)l1j(2+ n)l2j (1+ 1)m1j(2+ 1)m2j zj j! =?n;l1,l2; n;m1,m2; ? z ? ? ? ? 1,2;1,0; 1+ 1,2+ 1;0; ? ,(1.8) sl,m n,n(z;,):= n n ? j=0 (n)nj(+ n)lj (+ 1)mj zj j! =?n;l; n;m; ? z ? ? ? ? ;1; + 1;0; ? ,(1.9) bn,l n,m(z;,):= n n ? j=0 (n)nj(+n)lj (+n)mj zj j! =?n;l; n;m; ? z ? ? ? ? ; ; ? ,(1.10) t n,l1,l2,l3 n,m1,m2,m3(z;1,2,3,1,2,3) := ?n n j=0 (n)nj(1+n)l1j(2n)l2j(3)l3j (1+n)m1j(2n)m2j(3)m3j zj j! =?n;l1,l2,l3; n;m1,m2,m3; ? z ? ? ?1,2,3;1,1,0; 1,2,3;1,1,0; ? ,(1.11) ?n;(k r);(lp) n;(ts);(mq) ? z ? ? ? ? 1,.,r;1,.,p 1,.,s;1,.,q ? := ?n n j=0 (n)nk(1+n)k1j.(r+n)krj(1)l1j.(p)lpj (1n)t1j.(sn)tsj(1)m1j.(q)mqj zj j! =?n;k 1,.,kr;l1,.,lp n;t1,.,ts;m1,.,mq ? z ? ? ? 1,.,r;1,.,1;1,.,p 1,.,s;1,.,1;1,.,q ? .(1.12) the main object of this paper is to obtain a multiple integral representations associated with the polynomials defi ned by (1.6). these multiple integral representations generalize and unify the numerous results given recently by several authors, see for example 3,4,1219. many special cases involving well-known families of polynomials are also given. as mentioned by srivastava et al. 14, each integral representations derived in this paper may be viewed as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials. s. gaboury, r. tremblay/applied mathematics and computation 261 (2015) 394741 2. generalized multiple integral representations for the polynomials?n;(kr);(lp) n;(ts);(mq) ?z;( r),(r),(p);(s),(s),(q) ? defi ned by (1.6), we consider the following product: ?n;(k r);(lp) m;(ts);(mq) ?x; (r),(r),(p);(s),(s),(q)?n;(k r);(lp) n;(ts);(mq) ?y; (r),(r),(p);(s),(s),(q)? = ?m n j=0 ?n n l=0 (m)nj(n)nl(1+1m)k1j.(r+rm)krj(1+1n)k1l.(r+rn)krl (1+1m)t1j.(s+sm)tsj(1+1n)t1l.(s+sn)tsl (1)l1j.(p)lpj(1)l1l.(p)lpl (1)m1j.(q)mqj(1)m1l.(q)mql xj j! yl l! = ?j+lm+n n j,l=0 (mn)(j+l)n p ? i=1 (i+i)(j+l)li r ? i=1 (i+i+im +in)(j+l)ki q ? i=1 (i+i 1)(j+l)mi s ? i=1 (i+i+im +in 1)(j+l)ti xj j! yl l! m! n! (m+n)! ?(m+n+1(j+l)n) ?(mnj+1)?(nnl+1) ?p i=1 ?(i+i) ?(i)?(i) ?q i=1 ?(i)?(i) ?(i+i1) ? ?r i=1 ?(i+im+kij)?(i+in+kil) ?(i+i+im+in+(j+l)ki) ?s i=1 ?(i+i1+im+in+(j+l)ti) ?(i+im+tij)?(i+in+til) ? ?r i=1 ?(i+i+im+in) ?(i+im)?(i+in) ?s i=1 ?(i+im)?(i+in) ?(i+i+im+in1) ?p i=1 ?(i+lij)?(i+lil) ?(i+i+(j+l)li) ? ?q i=1 ?(i+i1+(j+l)mi) ?(i+mij)?(i+mil) ? ,(2.1) where we used repeatedly the following identity: (n)nk= (1)nkn! (n nk)! ? 0 k ?n n ? ;n n;n n0 ? .(2.2) let us now recall the well-known integral formula (see 20, p. 263, example 39 and also 21, p. 9) ? /2 0 cos+cos ?( )?d= 2+1 ?(+ 1) ?(+ 1)?(+ 1) ?re( +) 1?,(2.3) which can be rewritten as: ?(+ 1) ?(+ 1)?(+ 1) = 2+ ? /2 /2 ei()cos+d ?re( +) 1?.(2.4) by making use of (2.4) appropriately as well as the celebrated euler beta integral in the form ? b a (t a)1(b t)1dt =(b a)+1 ?()?() ?(+) ?minre(),re() 0;a ? = b?,(2.5) we fi nd from (2.1) ?n;(k r);(lp) m;(ts);(mq) ?x; (r),(r),(p);(s),(s),(q)?n;(k r);(lp) n;(ts);(mq) ?y; (r),(r),(p);(s),(s),(q)? = 2m+n+ ?s i=1(i+i+im+in2)+ ?q i=1(i+i2) q+s+1(ba)pr+ ?r i=1(i+i+im+in)+ ?p i=1(i+i) ?p i=1 ?(i+i) ?(i)?(i) ?q i=1 ?(i)?(i) ?(i+i1) ? m! n! (m+n)! ?r i=1 ?(i+i+im+in) ?(i+im)?(i+in) ?s i=1 ?(i+im)?(i+in) ?(i+i+im+in1) ? ? 2 2 d ? 2 2 d1? 2 2 ds ? 2 2 d1? 2 2 dq ?b a d1? b a dr ?b a d1? b a dp ?r i=1(i a) i+im1(b i)i+in1?p i=1(i a) i1(b i)i1? ?j+lm+n n j,l=0 (mn)(j+l)n p ? i=1 (i+i)(j+l)li r ? i=1 (i+i+im +in)(j+l)ki q ? i=1 (i+i 1)(j+l)mi s ? i=1 (i+i+im +in 1)(j+l)ti ?( s),(s),(s),(s),(q),(q) m,n (,(s),(q)(x;,(s),(q),(r),(p) j j! (y;,(s),(q),(r),(p)l l! (2.6) 42s. gaboury, r. tremblay/applied mathematics and computation 261 (2015) 3947 ? min 1ipre(i),re(i) 0; min 1irre(i +im),re(i+in) 0; min 1iqre(i +i) 1; min 1isre(i +i+im +in) 1;a ? = b;(kr),(lp),(ts),(mq) n 0;n n;(r),(r),(p);(s),(s),(q) c ? , where we set, for convenience, ?( s),(s),(s),(s),(q),(q) m,n (,(s),(q):= ei(m+n)+ ?s i=1(ii+imin)i+ ?q i=1(ii)i)cosm+n ?s i=1cos i+i+im+in2i?q i=1cos i+i2i?, (2.7) (x;,(s),(q),(r),(p):= ?2cos ?n ?s i=1 ?2cos i?t i?q i=1 ?2cos i?m i? ? r i=1 ? ia ba ?ki? ? p i=1 ? ia ba ?li? x ei( ?s i=1tii+ ?q i=1miin) (2.8) and (y;,(s),(q),(r),(p):= ?2cos ?n ?s i=1 ?2cos i?t i?q i=1 ?2cos i?m i? ? r i=1 ? bi ba ?ki? ? p i=1 ? bi ba ?li? y ei( ?s i=1tii+ ?q i=1miin). (2.9) making use of the following series identity 22, p. 52, equation (1.6) which holds in the case that each of the series involved is absolutely convergent: ? m,n=0 f(m + n)x m m! yn n! = ? n=0 f(n)(x + y) n n! ,(2.10) the double sum appearing in (2.6) can be reduced to a single sum which is interpreted by means of defi nition (1.6). we thus obtain the following multiple integral representations for the product of two polynomials of the generalized class defi ned by (1.6): ?n;(k r);(lp) m;(ts);(mq) ?x; (r),(r),(p);(s),(s),(q)?n;(k r);(lp) n;(ts);(mq) ?y; (r),(r),(p);(s),(s),(q)? = 2m+n+ ?s i=1(i+i+im+in2)+ ?q i=1(i+i2) q+s+1(b a)pr+ ?r i=1(i+i+im+in)+ ?p i=1(i+i) ? p ? i=1 ?(i+i) ?(i)?(i) ? q ? i=1 ?(i)?(i) ?(i+i 1) ? m! n! (m + n)! ? r ? i=1 ?(i+i+im +in) ?(i+im)?(i+in) ? s ? i=1 ?(i+im)?(i+in) ?(i+i+im +in 1) ? ? 2 2 d ? 2 2 d1 ? 2 2 ds ? 2 2 d1 ? 2 2 dq ? b a d1 ? b a dr ? b a d1 ? b a dp ? r ? i=1 (i a)i+im1(b i)i+in1 ? p ? i=1 (i a)i1(b i)i1 ? ?( s),(s),(s),(s),(q),(q) m,n (,(s),(q) ?n;(k r);(lp) m+n;(ts);(mq) ? ?(ts);(mq),n ? x ? r ? i=1 ? i a b a ?ki?p ? i=1 ? i a b a ?li? ,y ? r ? i=1 ?b i b a ?ki?p ? i=1 ?b i b a ?li? ;,(s),(q) ? ;(2.11) (r+r), ? rm +rn m + n ? ,(p+p);(s+s 1), ? sm +sn m + n ? ,(q+q 1) ? (2.12) (min 1ipre(i),re(i) 0; min 1irre(i +im),re(i+in) 0; min 1iqre(i +i) 1; min 1isre(i +i+im +in) 1; a ? = b;(kr),(lp),(ts),(mq) n 0;n n;(r),(r),(p),(s),(s),(q),(r),(r),(p),(s),(s),(q) c), where (p+p) abbreviates the array of p parameters 1+1,.,p+p with similar interpretations for (r+r),(s+s 1)and(q+q 1). s. gaboury, r. tremblay/applied mathematics and computation 261 (2015) 394743 the expression?(s),(s),(s),(s),(q),(q) m,n (,(s),(q)is given by (2.7) and ?(ts);(mq),nx,y;,(s),(q):=(2cos)n ? s i=1(2cosi) ti?q i=1(2cosi) mi? ? x ei( ?s i=1tii+ ?q i=1miin)+ y ei( ?s i=1tii+ ?q i=1miin) ? .(2.13) 3. applications to hypergeometric polynomials in this section, we rewrite the polynomials?n;(kr);(lp) m;(ts);(mq) ?x;( r),(r),(p);(s),(s),(q) ? as generalized hypergeometric polynomials. to this end, we use the next formula: ()nk= nnk n ? j=1 ? + j 1 n ? k (n n;k n0; c).(3.1) we thus obtain the following relation: ?n;(k r);(lp) m;(ts);(mq) ?x; (r),(r),(p);(s),(s),(q)?= n+k1+kr+l1+lpft1+ts+m1+mq ? ?(n;n),?(k1;1+1n),.,?(kr;r+rn),?(l1;1),.,?(lp;p); ?(t1;1+1n),.,?(ts;s+sn),?(m1;1),.,?(mq;q); ? nnkk1 1 .kkr rl l1 1 .l lp p tt1 1 .tts sm m1 1 .m mq q ? x ? ,(3.2) where?(n;) abbreviates the array of n parameters n , + 1 n ,., + n 1 n ,(n n) andpfq denotes the generalized hypergeometric function with p numerator and q denominator parameters, defi ned by 23, chapter 4 pfq(1,.,p;1,.,q;z)=pfq 1, .,p; z 1, .,q; = ? n=0 (1)n.(p)n (1)n.(q)n zn n! (3.3) ?p,q n 0;p q + 1;p q and|z| 0; min 1irre(i +im),re(i+in) 0; min 1iqre(i +i) 1; min 1isre(i +i+im +in) 1; a ? = b;(kr),(lp),(ts),(mq) n 0;n n;(r),(r),(p),(s),(s),(q),(r),(r),(p),(s),(s),(q) c?, and the expressions ?( s),(s),(s),(s),(q),(q) m,n (,(s),(q)and?(ts);(mq),nx,y;,(s),(q) are given by (2.7) and (2.13) respectively. 4. further special cases inthissection,wegivesomespecialcasesofeq.(3.4).thesespecialcasesinvolvevariouswell-knownfamiliesofpolynomials. we fi rst begin by computing the integral representation of the konhauser polynomials 24, p. 304, eq. (5) defi ned as follows: z n(x;k)= ?(kn + 1) n! n ? j=0 (1)j ?n j ? xkj ?(kj + 1) = (+ 1)kn n! 1fk n; ?x k ?k ?(k;+ 1) (k n).(4.1) letting n = 1, k1= k2= ? = kr= l1= l2= ? = lp= 0, t1= k, t2= t3= ? = ts= m1= m2= ? = mq= 0,1? + 1,1= 0,1? + 1,1= 0, x ? ? x k ?k and y ? ?y k ?k in (3.4), we obtain 1fk m; ?x k ?k ?(k;+ 1) 1fk n; ?y k ?k ?(k;+ 1) = 2m+n+ m! n! (m+n)! ?(+1)?(+1) ?(+1) ? 2 2 d ? 2 2 d1?(+1,+1,(0),(0),(0),(0) m,n (,1,(0)1fk m n; ?1;(0),1 ? x k ?k , ?y k ?k ;,1,(0) ? ?(k;+ 1); (4.2) ?re( +) 1;, c;k n;m,n n0?, where ?(+1,+1,(0),(0)(0),(0),(0) m,n (,1,(0)= ei(m+n)+()1)cosm+ncos+1(4.3) and ?1;(0),1 ? x k ?k , ?y k ?k ;,1,(0) ? = cos1 cos ? x k ?k ei(1)+ ?y k ?k ei(1) ? .(4.4) by making use of assertions (4.1) and (4.2), we obtain the following integral representation for the konhauser polynomials z n(x;k); z m(x;k)z n(y;k)= 2m+n+ (+1)km(+1)kn?(+1)?(+1) ?(+1) ? 2 2 d ? 2 2 d1?(+1,+1,(0),(0),(0),(0) m,n (,1,(0) z+ m+n ? k ? ?1;(0),1 ? x k ?k , ?y k ?k ;,1,(0) ?1 k ;k ? .(4.5) now let us shift our focus on an application of eq. (3.4) to the jacobi polynomials which are given in the next form 25: p(,) n (x)= ? + 2n n ?x + 1 2 ?n 2f1 n, n ; 2 x + 1 2n ; . (4.6) setting n = 1, k1= 1, k2= k3= ? = kr= l1= l2= ? = lp= 0 (r = 1, p = 0), t1= 1, t2= t3= ? = ts= m1= m2= ? = mq= 0(s = 1, q = 0),1? ,1= 1,1? ,1= 2,1? ,1= 1,1? and1= 2 in eq. (3.4) yields 2f1 m, m; x 2m; 2f1 n, n; y 2n; = 2mn 2(ba)1mn m! n! (m+n)! ?(mn) ?(m)?(n) ?(2m)?(2n) ?(2m2n1) s. gaboury, r. tremblay/applied mathematics and computation 261 (2015) 394745 ? 2 2 d ? 2 2 d1 ?b a d1?(,2,2,(0),(0) m,n (,1,(0)(1 a)m1 ?b 1? n1 (4.7) 2f1 m n, (m + n); ?1;(0),1 ? x ? 1 a b a ? ,y ?b 1 b a ? ;,1,(0) ? 1 2(m + n); (4.8) ?minre( m),re( n) 0;re( 1) 2(m + n);a ? = b;, c?, where ?(,2,2,(0),(0) m,n (,1,(0)= ei(m+n)+(2m2n)1)cosm+ncos2mn2n21(4.9) and ?1;(0),1 ? x ? 1 a b a ? ,y ?b 1 b a ? ;,1,(0) ? = cos1 cos ? x ? 1 a b a ? ei(1)+ y ?b 1 b a ? ei(1) ? .(4.10) replacing x by 2 1+x and y by 2 1+y in (4.7) and with the help of assertion (4.6), we have the following integral representation for the product of jacobi polynomials: p(,) m (x)p(,) n (y)= 2mn 2(ba)1mn ?+2m m ?+2n n ?x+1 2 ?m? y+1 2 ?n ?(2m)?(2n) ?(2m2n1) ?(mn) ?(m)?(n) ? 2 2 d ? 2 2 d1 ?b a d1?(,2,2,(0),(0) m,n (,1,(0)(1 a)m1 ?b 1? n1 (4.11) m! n! (m+n)! ? ?1;(0),1 ? 2 1 + x ? 1 a b a ? , 2 1 + y ?b 1 b a ? ;,1,(0) ?m+n? + 1 + 2(m + n) m + n ?1 p(+1,+) m+n 2?1;(0),1 2 1 + x ? 1 a b a ? , 2 1 + y ?b 1 b a ? ;,1,(0) ?1;(0),1 2 1 + x ? 1 a b a ? , 2 1 + y ?b 1 b a ? ;,1,(0) . (4.12) finally, we give a last special case that involves the generalized sister celine s polynomials investigated by khan and shukla 26 and defi ned as follows: fn(k,;1,.,p;1,.,q;x)=p+k+1fq+k+1 ?(k;n);n +,(p); x ?(k + 1;);(q); . (4.13) putting k1= 1, k2= k3= ? = kr= 0(r = 1), t1= n + 1, t2