Thông tin tài liệu
He et al Advances in Difference Equations (2016) 2016:287 DOI 10.1186/s13662-016-1014-0 RESEARCH Open Access Some new formulas for the products of the Apostol type polynomials Yuan He1 , Serkan Araci2* and HM Srivastava3,4 * Correspondence: mtsrkn@hotmail.com Department of Economics, Faculty of Economics, Administrative and Social Sciences, Hasan Kalyoncu University, Gaziantep, 27410, Turkey Full list of author information is available at the end of the article Abstract In the year 2014, Kim et al computed a kind of new sums of the products of an arbitrary number of the classical Bernoulli and Euler polynomials by using the Euler basis for the vector space of polynomials of bounded degree Inspired by their work, in this paper, we establish some new formulas for such a kind of sums of the products of an arbitrary number of the Apostol-Bernoulli, Euler, and Genocchi polynomials by making use of the generating function methods and summation transform techniques The results derived here are generalizations of the corresponding known formulas involving the classical Bernoulli, Euler, and Genocchi polynomials MSC: Primary 11B68; secondary 05A19 Keywords: Apostol-Bernoulli polynomials; Apostol-Euler polynomials; Apostol-Genocchi polynomials; summation formulas; recurrence relations Introduction The classical Bernoulli polynomials Bn (x), Euler polynomials En (x), and Genocchi polynomials Gn (x) are usually defined by the following generating functions: text = et – ext = et + ∞ Bn (x) tn n! |t| < π , (.) En (x) tn n! |t| < π , (.) tn n! |t| < π (.) n= ∞ n= and text = et + ∞ Gn (x) n= The rational numbers Bn , the integers En , and the rational numbers Gn given by Bn = Bn (), En = n En , and Gn = Gn () are called the classical Bernoulli numbers, Euler numbers, and Genocchi numbers, respectively These polynomials and numbers play important roles in many different areas © He et al 2016 This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made He et al Advances in Difference Equations (2016) 2016:287 Page of 18 of mathematics, such as number theory, combinatorics, special functions and analysis Numerous interesting properties for them can be found in many books and papers (see, for example, [–]) Some widely investigated analogs of the above classical Bernoulli, Euler and Genocchi polynomials are the Apostol-Bernoulli polynomials Bn (x; λ), Apostol-Euler polynomials En (x; λ) and Apostol-Genocchi polynomials Gn (x; λ), which are usually defined by means of the following generating functions (see, e.g., [–]): text = λet – ∞ Bn (x; λ) n= tn n! |t| < π when λ = ; |t| < | log λ| when λ = , ext = λet + ∞ En (x; λ) n= (.) tn n! |t| < π when λ = ; |t| < log(–λ) when λ = , (.) and text = λet + ∞ Gn (x; λ) n= tn n! |t| < π when λ = ; |t| < log(–λ) when λ = (.) In particular, Bn (λ), En (λ), and Gn (λ) given by Bn (λ) = Bn (; λ), En (λ) = n En ;λ , and Gn (λ) = Gn (; λ) are called the Apostol-Bernoulli numbers, Apostol-Euler numbers, and Apostol-Genocchi numbers, respectively Obviously, Bn (x; λ), En (x; λ), and Gn (x; λ) reduce, respectively, to Bn (x), En (x), and Gn (x) when λ = It is worth mentioning that the Apostol-Bernoulli polynomials were first introduced by Apostol [] (see also Srivastava [] for a systematic further study) in order to evaluate the value of the Hurwitz-Lerch zeta function Since the publication of the work by Luo and Srivastava [–], some interesting properties for the Apostol-Bernoulli, Euler and Genocchi polynomials have been well explored by many authors (see, for example, [–]) The present paper is concerned with the sums of the products of an arbitrary number of the above-mentioned polynomials and numbers The best known such formula is Dilcher’s result on the following sums of the products of an arbitrary number of the classical Bernoulli polynomials (see, for details, []): i +···+ik =n (i , ,ik ) n Bi (x ) · · · Bik (xk ) i , , ik = (–)k– k n k k– i (–)i i= j= k–i–+j Bn–i (y) , s(k, k – i + j)yj n–i j (.) He et al Advances in Difference Equations (2016) 2016:287 Page of 18 where n and k are positive integers (with n cients given by k), n i , ,ik denotes the multinomial coeffi- n! n = , i ! · · · ik ! i , , ik (.) s(n, k) are the Stirling numbers of the first kind and y = x + · · · + xk We refer to [–] for some extensions of (.) in different directions In the year , Kim et al [] considered and computed the following kind of new sums of the products of an arbitrary number of the classical Bernoulli and Euler polynomials by making use of the Euler basis for the vector space of polynomials of bounded degree: s r Vn;r,s (x) = Bik (x) Ir +Js =n k= = n– k= Ejk (x) k= n+r+s– αn,k (r, s)Ek (x) k n+r+s– En (x), n + (.) where n, r, and s are positive integers, (.) Ir +Js =n denotes the sum over all non-negative integers i , , ir and j , , js such that i + · · · + ir + j + · · · + js = n, and αn,k (r, s) is a rational number determined by s r αn,k (r, s) = j= i=max(,r+k–n) + Vn–k;r,s () r i s (–)j s–j Vn+i–k–r;i,j () j (.) Motivated and inspired by the work of Kim et al [], in this paper, we establish some new formulas for such a kind of sums of the products of an arbitrary number of the Apostol-Bernoulli, Euler and Genocchi polynomials by making use of the generating function methods and summation transform techniques As applications, some known results for the classical Bernoulli, Euler, and Genocchi polynomials are shown to be derivable as special cases of our product formulas Our paper is organized as follows In Section , we give several new formulas for the products of the Apostol-Bernoulli, Euler, and Genocchi polynomials Various corollaries and consequences of these main results are also considered in Section itself Section is devoted to the proofs of the main results He et al Advances in Difference Equations (2016) 2016:287 Page of 18 Statements of the main results Let r and s be positive integers and let λ , , λr and μ , , μs be r + s parameters For convenience, in the following, we always denote by λ a parameter given by r s λ= μk , λk k= (.) k= with Ir +Js =n the same as in (.), and by Ma , Nb , and Tb three sequences of polynomials given (for positive integers a and b) with a– r λk Bik (xk – xa + ; λk ) Ma = k= Bik (xk – xa ; λk ), (.) Ejk (yk – yb ; μk ), (.) Gjk (yk – yb ; μk ), (.) k=a+ b– s μk Ejk (yk – yb + ; μk ) Nb = k= k=b+ and b– s μk Gjk (yk – yb + ; μk ) Tb = k= k=b+ respectively We also write, for subsets R ⊆ {, , r} and S ⊆ {, , s}, |R| as the cardinality of R and |S| as the cardinality of S, R = {, , r}\R and S = {, , s}\S for positive integers r and s In particular, if |R| = a and |S| = b for positive integers a and b, we denote s , , sr–a ∈ R and r , , rs–b ∈ S We now state our results as follows Theorem Let r and s be positive integers Also let s be an even integer Then, for every non-negative integer n, s r Bik (xk ; λk ) (n + r + s) Ir +Js =n k= r = Ir +Js =n a= Ejk (yk ; μk ) k= n+r+s Bia (xa ; λ)Ma ia s + Ir +Js =n+ b= s Ejk (yk – xa ; μk ) k= n+r+s (–)b Bjb (yb ; λ)Nb jb r λk Bik (xk – yb + ; λk ) k= (.) He et al Advances in Difference Equations (2016) 2016:287 Page of 18 Furthermore, if s is an odd positive integer, then, for every positive integer n, s r Bik (xk ; λk ) Ir +Js =n k= =– I +J =n– r Ejk (yk ; μk ) k= s r a= s n+r+s– Eia (xa ; λ)Ma ia s Ejk (yk – xa ; μk ) k= n+r+s– (–)b Ejb (yb ; λ)Nb jb – Ir +Js =n b= r λk Bik (xk – yb + ; λk ) (.) k= We now deduce some special cases of Theorem Since the Apostol-Bernoulli and Apostol-Euler polynomials satisfy the following difference equations (see, e.g., []): λBn (x + , λ) – Bn (x, λ) = nxn– (n ) (.) and λEn (x + , λ) + En (x, λ) = xn (n ), (.) respectively, so we find from (.) and (.) that a– λk Bik (xk – xa + , λk ) k= Bik (xk – xa , λk ) = T⊆{, ,a–} k∈T ik (xk – xa )ik – (.) k∈T and b– –μk Ejk (yk – yb + , μk ) k= Ejk (yk – yb , μk ) = T⊆{, ,b–} k∈T –(yk – yb )jk (.) k∈T Hence, by setting x = · · · = xr = x and y = · · · = ys = y in Theorem , in view of (.) and (.), we obtain the following result Corollary Let r and s be positive integers Also let s be an even integer Then, for every non-negative integer n, s r Bik (x; λk ) (n + r + s) Ir +Js =n k= r = a= |R|=a Ir–a +i +Js =n+–a Ejk (y; μk ) k= n+r+s Bi (x; λ) i He et al Advances in Difference Equations (2016) 2016:287 r–a Page of 18 s Bik (λsk ) · k= Ejk (y – x; μk ) k= s n+r+s Bj (y; λ) j (–)b + b= |S|=b Ir +j +Js–b =n+ s–b r Ejk (; μrk ) · k= λk Bik (x – y + ; λk ) (.) k= Moreover, if s is an odd positive integer, then, for every positive integer n, s r Bik (x; λk ) Ir +Js =n k= Ejk (y; μk ) k= r =– a= |R|=a Ir–a +i +Js =n–a r–a s Bik (λsk ) · n+r+s– Ei (x; λ) i k= Ejk (y – x; μk ) k= s n+r+s– Ej (y; λ) j (–)b– + b= |S|=b Ir +j +Js–b =n s–b r Ejk (; μrk ) · k= λk Bik (x – y + ; λk ) (.) k= Since the Apostol-Bernoulli polynomials satisfy the following symmetric distribution (see, e.g., []): λBn ( – x; λ) = (–)n Bn x; λ (n ), (.) by setting λ = · · · = λr = and μ = · · · = μs = in Corollary , we get the following formulas for the products of an arbitrary number of the classical Bernoulli polynomials and the classical Euler polynomials Corollary Let r and s be positive integers If s is an even positive integer, then, for every non-negative integer n, s r Bik (x) (n + r + s) Ir +Js =n k= r r a = a= s + b= Ir–a +i +Js =n+–a r +Js–b =n+ r–a s Ejk (y – x) Bik k= k= n+r+s (–)Ir Bj (y) j r Ejk () k= n+r+s Bi (x) i s (–)b b I +j s–b · Ejk (y) k= Bik (y – x) k= (.) He et al Advances in Difference Equations (2016) 2016:287 Page of 18 Furthermore, if s is an odd positive integer, then, for every positive integer n, s r Bik (x) Ir +Js =n k= =– Ejk (y) k= r a= r–a · r a Ir–a +i +Js =n–a s Ejk (y – x) Bik k= k= s + b= s (–)b– b I +j r s–b · n+r+s– Ei (x) i +Js–b =n n+r+s– (–)Ir Ej (y) j r Ejk () k= Bik (y – x) (.) k= In the special case when x = y, Corollary yields the corresponding new expressions for the above-mentioned sums of the products of an arbitrary number of the classical Bernoulli polynomials and the classical Euler polynomials considered by Kim et al [] If we take r = s = in Corollary , in light of (.), we obtain the following result Corollary Let n be a positive integer Then n Bk (x; λ)En–k (y; μ) k= =– n– k= n + k= n+ En––k (x; λμ)Ek (y – x; μ) k+ n+ En–k (y; λμ) Bk (x – y; λ) + k(x – y)k– k+ (.) In particular, since (see, e.g., []) En () = – n+ Bn+ n+ (n ), by setting x = y and λ=μ= in Corollary , we find for every positive integer n n n Bk (x)En–k (x) – k= = (n + )En (x), k= n+ k+ k + k – that Bk En–k (x) k (.) which was derived by Pan and Sun [] by using the finite difference calculus and differentiation He et al Advances in Difference Equations (2016) 2016:287 Page of 18 Theorem Let r and s be positive integers Then, for every non-negative integer n, s r Bik (xk ; λk ) (n + r + s) Ir +Js =n k= r = Ir +Js =n a= Gjk (yk ; μk ) k= n+r+s Pia (xa ; λ)Ma ia s s Gjk (yk – xa ; μk ) k= n+r+s (–)b Pjb (yb ; λ)Tb jb + Ir +Js =n b= r λk Bik (xk – yb + ; λk ) , · (.) k= where Pn (x; λ) is given by ⎧ ⎨B (x; λ) ( | s), n Pn (x; λ) = ⎩– Gn (x; λ) ( s) We now deduce some special cases of Theorem Since the Apostol-Genocchi polynomials satisfy the following difference equation (see, e.g., []): λGn (x + , λ) + Gn (x, λ) = nxn– (n ), (.) by applying (.), we have b– –μk Gjk (yk – yb + , μk ) k= Gjk (yk – yb , μk ) = T⊆{, ,b–} k∈T –jk (yk – yb )jk – k∈T Hence, by setting x = · · · = xr = x and y = · · · = ys = y in Theorem , and in view of (.) and (.), we obtain the following result Corollary Let r and s be positive integers Then, for every non-negative integer n, s r Bik (x; λk ) (n + r + s) Ir +Js =n k= r = a= |R|=a Ir–a +i +Js =n+–a r–a k= n+r+s Pi (x; λ) i s Bik (λsk ) · Gjk (y; μk ) k= Gjk (y – x; μk ) k= (.) He et al Advances in Difference Equations (2016) 2016:287 Page of 18 s (–)b + b= |S|=b Ir +j +Js–b =n+–b s–b r Gjk (μrk ) · n+r+s Pj (y; λ) j k= λk Bik (x – y + ; λk ) (.) k= Upon setting λ = · · · = λr = and μ = · · · = μs = in Corollary , if we make use of (.), we obtain the following formula for the products of an arbitrary number of the classical Bernoulli and Genocchi polynomials Corollary Let r and s be positive integers Then, for every non-negative integer n, s r (n + r + s) Bik (x) Ir +Js =n k= r r a = a= s + b= Ir–a +i +Js =n+–a s (–)b b s–b · Gjk (y) k= n+r+s Pi (x) i Ir +j +Js–b =n+–b r–a s Gjk (y – x) Bik k= k= n+r+s (–)Ir Pj (y) j r Bik (y – x), Gjk k= (.) k= where Pn (x) is given by ⎧ ⎨B (x) ( | s), n Pn (x) = ⎩– Gn (x) ( s) If we take r = s = in Corollary , in light of (.), we get the following result Corollary Let n be a non-negative integer Then n Bk (x; λ)Gn–k (y; μ) k= =– n k= n + k= n+ k k– n+ k k– Gk (x; λμ)Gn–k (y – x; μ) Gk (y; λμ) Bn–k (x – y; λ) + (n – k)(x – y)n–k– (.) Since the classical Genocchi polynomials can be expressed in terms of the classical Bernoulli polynomials as follows: Gn (x) = Bn (x) – n+ Bn x (n ), (.) He et al Advances in Difference Equations (2016) 2016:287 Page 10 of 18 by setting λ = μ = and x = y in Corollary , and in light of the fact that (see, e.g., [, ]) B = , B = – and G (x) = , we find for every positive integer n n– n– Bk (x)Gn–k (x) – k= k= n+ k k– that n–k Gk (x)Bn–k = (n – )Gn (x), (.) which was derived by Agoh [] by applying some short and intelligible ideas For some convolution formulas similar to (.) and (.), the interested reader may be referred to [–] Proofs of Theorems and In our proofs of Theorems and , we need the following auxiliary result described in [, ] Lemma Let n be a positive integer with n the standard simplex in Rn ) defined by n := (t , , tn ) : tk and let n (k = , , n) and t + · · · + tn be the n-dimensional space (or Then the multivariable Beta function B(α , , αn ) is given by the following Dirichlet integral: B(α , , αn ) = = (α ) · · · (αn ) (α + · · · + αn ) n– tα – · · · tn– α ··· – n– · ( – t – · · · – tn– )αn – dt · · · dtn– (α ), , (αn ) > (.) Proof of Theorem We first recall the following elementary and beautiful idea: ( + x )( + x )( + x ) · · · = ( + x ) + x ( + x ) + x ( + x )( + x ) + · · · , (.) which was used by Euler to give the proof of his famous pentagonal number theorem (see, e.g., [, ]) Obviously, the finite form of (.) can be expressed as follows: ( + x ) · · · ( + xn ) = ( + x ) + x ( + x ) + · · · + xn ( + x ) · · · ( + xn– ) (.) He et al Advances in Difference Equations (2016) 2016:287 For Page 11 of 18 n, if we write xk – for xk in (.), we get k n x · · · xn – = (xk – )x · · · xk– , (.) k= where the product x · · · xk– is assumed to be equal to when k = Let εk be a piecewise function of k given by εk = ⎧ ⎨λ ( k ⎩–μk–r k r), (r + (.) r + s) k By replacing n by r + s and taking xk = εk etk in (.), we find that (–)s λet +···+tr+s – r k– εk etk – = s εi eti + i= k= r+k– εr+k etr+k – εi eti , (.) i= k= which, together with (.), yields (–)s λet +···+tr+s – r a– ta = λa e – a= s ti λi e + i= b– b tr+b (–) μb e + λi eti μi e i= b= r tr+i (.) i= It follows from (.) that r k= tk exk tk λk etk – = s k= eyk tr+k μk etr+k + r (–)s λet +···+tr+s – r · k= s tk exk tk λk etk – k= a– λa eta – a= i= eyk tr+k μk etr+k + s b– (–)b μb etr+b + + r (xk +)tk λk k= μi etr+i i= b= · λi eti tk e λk etk – s k= eyk tr+k μk etr+k + (.) We now observe that a– λa eta – r λi eti i= k= a– = ta exa (t +···+tr ) λk k= tk exk tk λk etk – tk e(xk –xa +)tk λk etk – r k=a+ tk e(xk –xa )tk λk etk – (.) He et al Advances in Difference Equations (2016) 2016:287 Page 12 of 18 and b– s μb etr+b + eyk tr+k μk etr+k + μi etr+i i= k= b– = eyb (tr+ +···+tr+s ) e(yk –yb +)tr+k μk etr+k + μk k= s k=b+ e(yk –yb )tr+k μk etr+k + (.) Thus, by applying (.) and (.) to (.), we obtain r k= tk exk tk λk etk – r s eyk tr+k μk etr+k + k= a– ta exa (t +···+tr+s ) (–)s λet +···+tr+s – = a= r · k=a+ tk e(xk –xa )tk λk etk – s (–)b + b= s · k=b+ s λk k= tk e(xk –xa +)tk λk etk – e(yk –xa )tr+k μk etr+k + k= eyb (t +···+tr+s ) (–)s λet +···+tr+s – r e(yk –yb )tr+k μk etr+k + λk k= b– μk k= e(yk –yb +)tr+k μk etr+k + tk e(xk –yb +)tk λk etk – (.) For convenience, let tn f (t) n! denote the coefficient of substitute uk t for tk with tn n! in the power-series expansion of f (t) For k r + s, if we u + · · · + ur+s = into both sides of (.), we find that r tn n! = k= uk texk uk t λk euk t – r tn n! a= r · k=a+ + tn n! s · k=b+ s k= eyk ur+k t μk eur+k t + a– ua texa t (–)s λet – uk te(xk –xa )uk t λk euk t – s (–)b b= λk k= s k= uk te(xk –xa +)uk t λk euk t – e(yk –xa )ur+k t μk eur+k t + eyb t (–)s λet – e(yk –yb )ur+k t μk eur+k t + r λk k= b– μk k= e(yk –yb +)ur+k t μk eur+k t + uk te(xk –yb +)uk t λk euk t – =: M + M (.) He et al Advances in Difference Equations (2016) 2016:287 Page 13 of 18 The left-hand side of (.) can easily be rewritten as follows: r tn n! k= uk texk uk t λk euk t – s k= eyk ur+k t μk eur+k t + i r Bik (xk ; λk ) = n! · Ir +Js =n k= j s ukk ik ! Ejk (yk ; μk ) k= k ur+k jk ! (.) Moreover, M and M on the right-hand side of (.) can be rewritten as follows: r Fia (xa ; λ) M = n! · Ir +Js =n+ a= ua ia ! i r Bik (xk – xa ; λk ) · k=a+ i a– λk Bik (xk – xa + ; λk ) k= j s ukk ik ! ukk ik ! Ejk (yk – xa ; μk ) k= k ur+k jk ! (.) and s (–)b Fjb (yb ; λ) M = · n! · Ir +Js =n++ b= j b– k ur+k jk ! μk Ejk (yk – yb + ; μk ) · k= r λk Bik (xk – yb + ; λk ) · i ukk k= ik ! ur+b jb ! j s Ejk (yk – yb ; μk ) k=b+ , k ur+k jk ! (.) where = ⎧ ⎨ (s = , , , ), ⎩– (s = , , , ), and Fn (x; λ) is determined by ⎧ ⎨B (x; λ) ( | s), n Fn (x; λ) = ⎩– En (x; λ) ( s) (.) It follows from (.) to (.) that i r Bik (xk ; λk ) Ir +Js =n k= ukk ik ! Ejk (yk ; μk ) k= r Fia (xa ; λ) = Ir +Js =n+ a= j s ua ia ! r Bik (xk – xa ; λk ) · k=a+ k ur+k jk ! i a– λk Bik (xk – xa + ; λk ) k= i ukk ik ! j s Ejk (yk – xa ; μk ) k= k ur+k jk ! ukk ik ! He et al Advances in Difference Equations (2016) 2016:287 Page 14 of 18 s (–)b Fjb (yb ; λ) + Ir +Js =n++ b= j b– μk Ejk (yk – yb + ; μk ) · k= r λk Bik (xk – yb + ; λk ) · ur+b jb ! k= k ur+k jk ! i ukk ik ! j s Ejk (yk – yb ; μk ) k=b+ k ur+k jk ! (.) We note that, for complex numbers α , , αr+s with (α ), , (αr+s ) > –, if we use Lemma , we find for u + · · · + ur+s = that r+s uα · · · uαr+s du · · · dur+s– ··· r+s– (α + ) · · · (αr+s + ) (α + · · · + αr+s + r + s) = (.) Consequently, by the following operation: ··· (· · · ) du · · · dur+s– r+s– applied to both sides of (.), and with the help of (.), we get (n + r + s – )! I +J =n r r = Ir +Js =n+ a= s s r Bik (xk ; λk ) k= Ejk (yk ; μk ) k= Fia (xa ; λ) ia ! · (n + – ia + r + s)! r a– λk Bik (xk – xa + ; λk ) k= s Bik (xk – xa ; λk ) · Ejk (yk – xa ; μk ) k=a+ k= s (–)b + Ir +Js =n++ b= Fjb (yb ; λ) jb ! · (n + – jb + r + s)! b– μk Ejk (yk – yb + ; μk ) · k= s r Ejk (yk – yb ; μk ) · k=b+ λk Bik (xk – yb + ; λk ), (.) k= which, together with (.), yields the desired results (.) and (.) This completes the proof of Theorem He et al Advances in Difference Equations (2016) 2016:287 Page 15 of 18 Proof of Theorem Let u , , ur+s be r + s variables with u + · · · + ur+s = For that s, if we substitute ur+k teyk ur+k t for eyk ur+k t in both sides of (.), we find k r tn n! = k= uk texk uk t λk euk t – r tn n! a= r k= k=a+ tn n! s · k=b+ ur+k teyk ur+k t μk eur+k t + a– ua texa t (–)s λet – (xk –xa )uk t · + s uk te λk euk t – s (–)b b= uk te(xk –xa +)uk t λk euk t – λk k= s ur+k te(yk –xa )ur+k t μk eur+k t + k= b– ur+b teyb t (–)s λet – μk k= r ur+k te(yk –yb )ur+k t μk eur+k t + λk k= ur+k te(yk –yb +)ur+k t μk eur+k t + uk te(xk –yb +)uk t λk euk t – = N + N , (.) say It is trivial to obtain r tn n! k= uk texk uk t λk euk t – s k= ur+k teyk ur+k t μk eur+k t + i r Bik (xk ; λk ) = n! · Ir +Js =n k= j s ukk ik ! Gjk (yk ; μk ) k= k ur+k , jk ! (.) and N and N in the right-hand side of (.) can be rewritten as r Pia (xa ; λ) N = n! · Ir +Js =n a= r Bik (xk – xa ; λk ) · k=a+ ua ia ! a– i s ukk ik ! i λk Bik (xk – xa + ; λk ) k= ukk ik ! j Gjk (yk – xa ; μk ) k= k ur+k jk ! (.) and s (–)b Pjb (yb ; λ) N = · n! · Ir +Js =n b= j b– μk Gjk (yk – yb + ; μk ) · k= r λk Bik (xk – yb + ; λk ) · k= k ur+k jk ! i ukk ik ! ur+b jb ! j s Gjk (yk – yb ; μk ) k=b+ k ur+k jk ! (.) He et al Advances in Difference Equations (2016) 2016:287 Page 16 of 18 It follows from (.)-(.) that i r Bik (xk ; λk ) Ir +Js =n k= Gjk (yk ; μk ) k= r Pia (xa ; λ) = j s ukk ik ! Ir +Js =n a= ua ia ! i a– λk Bik (xk – xa + ; λk ) k= i r Bik (xk – xa ; λk ) · k=a+ ukk ik ! Gjk (yk – xa ; μk ) k= Ir +Js =n b= j b– μk Gjk (yk – yb + ; μk ) k= r λk Bik (xk – yb + ; λk ) · ··· By making the operation of (.), we get (n + r + s – )! I +J =n r r = Ir +Js =n a= s k ur+k jk ! i ukk k= ik ! r+s– k ur+k jk ! ur+b jb ! (–)b Pjb (yb ; λ) · ukk ik ! j s s + k ur+k jk ! j s Gjk (yk – yb ; μk ) k=b+ k ur+k jk ! (.) · du · · · dur+s– in both sides of (.), with the help s r Bik (xk ; λk ) k= Gjk (yk ; μk ) k= Pia (xa ; λ) ia ! · (n – ia + r + s)! r a– λk Bik (xk – xa + ; λk ) k= s Bik (xk – xa ; λk ) · k=a+ Gjk (yk – xa ; μk ) k= s (–)b + Ir +Js =n b= Pjb (yb ; λ) jb ! · (n – jb + r + s)! b– μk Gjk (yk – yb + ; μk ) · k= s r Gjk (yk – yb ; μk ) · k=b+ λk Bik (xk – yb + ; λk ), (.) k= as desired This concludes the proof of Theorem Competing interests The authors declare that they have no competing interests Authors’ contributions All authors participated in drafting, revising, and commenting on the manuscript All authors read and approved the final manuscript Author details Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan 650500, People’s Republic of China Department of Economics, Faculty of Economics, Administrative and Social Sciences, He et al Advances in Difference Equations (2016) 2016:287 Page 17 of 18 Hasan Kalyoncu University, Gaziantep, 27410, Turkey Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada China Medical University, Taichung, 40402, Taiwan, Republic of China Acknowledgements We express our sincere thanks to the anonymous referees for their comments on this manuscript This work was supported by the Foundation for Fostering Talents in Kunming University of Science and Technology (Grant No KKSY201307047) and the National Natural Science Foundation of the People’s Republic of China (Grant No 11326050) Received: 27 September 2016 Accepted: 27 October 2016 References Agoh, T, Dilcher, K: Convolution identities and lacunary recurrences for Bernoulli numbers J Number Theory 124, 105-122 (2007) Araci, S: Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus Appl Math Comput 233, 599-607 (2014) Cohen, H: Number Theory - Volume II: Analytic and Modern Tools Graduate Texts in Mathematics, vol 240 Springer, Berlin (2007) Nielsen, N: Traité élémentaire des nombres de Bernoulli Gauthier-Villars, Paris (1923) Srivastava, HM, Choi, J: Zeta and q-Zeta Functions and Associated Series and Integrals Elsevier, Amsterdam (2012) Srivastava, HM, Pintér, Á: Remarks on some relationships between the Bernoulli and Euler polynomials Appl Math Lett 17, 375-380 (2004) Luo, Q-M: Extension for the Genocchi polynomials and its Fourier expansions and integral representations Osaka J Math 48, 291-309 (2011) Luo, Q-M, Srivastava, HM: Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials J Math Anal Appl 308, 290-302 (2005) Luo, Q-M, Srivastava, HM: Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials Comput Math Appl 51, 631-642 (2006) 10 Apostol, TM: On the Lerch zeta function Pac J Math 1, 161-167 (1951) 11 Srivastava, HM: Some formulas for the Bernoulli and Euler polynomials at rational arguments Math Proc Camb Philos Soc 129, 77-84 (2000) 12 Lu, D-Q, Srivastava, HM: Some series identities involving the generalized Apostol type and related polynomials Comput Math Appl 62, 3591-3602 (2011) 13 Luo, Q-M: Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials Math Comput 78, 2193-2208 (2009) 14 Luo, Q-M, Srivastava, HM: Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind Appl Math Comput 217, 5702-5728 (2011) 15 Navas, LM, Ruiz, FJ, Varona, JL: Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials Math Comput 81, 1707-1722 (2011) 16 Prévost, M: Padé approximation and Apostol-Bernoulli and Apostol-Euler polynomials J Comput Appl Math 233, 3005-3017 (2010) 17 Srivastava, HM: Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials Appl Math Inf Sci 5, 390-444 (2011) 18 Dilcher, K: Sums of products of Bernoulli numbers J Number Theory 60, 23-41 (1996) 19 Chen, K-W: Sums of products of generalized Bernoulli polynomials Pac J Math 208, 39-52 (2003) 20 He, Y, Araci, S: Sums of products of Apostol-Bernoulli and Apostol-Euler polynomials Adv Differ Equ 2014, Article ID 155 (2014) 21 Huang, I-C, Huang, S-Y: Bernoulli numbers and polynomials via residues J Number Theory 76, 178-193 (1999) 22 Kim, M-S, Hu, S: Sums of products of Apostol-Bernoulli numbers Ramanujan J 28, 113-123 (2012) 23 Kim, T: Sums of products of q-Bernoulli numbers Arch Math 76, 190-195 (2001) 24 Kim, T, Adiga, C: Sums of products of generalized Bernoulli numbers Int Math J 5, 1-7 (2004) 25 Simsek, Y, Kurt, V, Kim, D: New approach to the complete sum of products of the twisted (h, q)-Bernoulli numbers and polynomials J Nonlinear Math Phys 14, 44-56 (2007) 26 Simsek, Y: Complete sum of products of (h, q)-extension of Euler polynomials and numbers J Differ Equ Appl 16, 1331-1348 (2010) 27 Kim, DS, Kim, T, Mansour, T: Euler basis and the product of several Bernoulli and Euler polynomials Adv Stud Contemp Math 24, 535-547 (2014) 28 Abramowitz, M, Stegun, IA (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables Applied Mathematics Series, vol 55 National Bureau of Standards, Washington (1964) Reprinted by Dover, New York, 1965 29 Pan, H, Sun, Z-W: New identities involving Bernoulli and Euler polynomials J Comb Theory, Ser A 113, 156-175 (2006) 30 Agoh, T: Convolution identities for Bernoulli and Genocchi polynomials Electron J Comb 21, Article ID P1.65 (2014) 31 Dunne, GV, Schubert, C: Bernoulli number identities from quantum field theory and topological string theory Commun Number Theory Phys 7, 225-249 (2013) 32 Faber, C, Pandharipande, R: Hodge integrals and Gromov-Witten theory Invent Math 139, 173-199 (2000) 33 He, Y, Zhang, WP: Some sum relations involving Bernoulli and Euler polynomials Integral Transforms Spec Funct 22, 207-215 (2011) 34 He, Y, Araci, S, Srivastava, HM, Acikgoz, M: Some new identities for the Apostol-Bernoulli polynomials and the Apostol-Genocchi polynomials Appl Math Comput 262, 31-41 (2015) 35 Kim, DS, Kim, T, Lee, S-H, Kim, Y-H: Some identities for the product of two Bernoulli and Euler polynomials Adv Differ Equ 2012, Article ID 95 (2012) 36 Kim, DS, Kim, T: Identities arising from higher-order Daehee polynomial bases Open Math 13, 196-208 (2015) He et al Advances in Difference Equations (2016) 2016:287 Page 18 of 18 37 Carlson, BC: Special Functions of Applied Mathematics Academic Press, New York (1977) 38 Srivastava, HM, Niukkanen, AW: Some Clebsch-Gordan type linearization relations and associated families of Dirichlet integrals Math Comput Model 37, 245-250 (2003) 39 Andrews, GE: Euler’s pentagonal number theorem Math Mag 56, 279-284 (1983) 40 Bell, J: A summary of Euler’s work on the pentagonal number theorem Arch Hist Exact Sci 64, 301-373 (2010) ... inspired by the work of Kim et al [], in this paper, we establish some new formulas for such a kind of sums of the products of an arbitrary number of the Apostol- Bernoulli, Euler and Genocchi polynomials. .. kind of new sums of the products of an arbitrary number of the classical Bernoulli and Euler polynomials by making use of the Euler basis for the vector space of polynomials of bounded degree:... special cases of our product formulas Our paper is organized as follows In Section , we give several new formulas for the products of the Apostol- Bernoulli, Euler, and Genocchi polynomials Various
Ngày đăng: 04/12/2022, 16:40
Xem thêm:
