Some formulae for products of Fubini polynomials with applications
aa r X i v : . [ m a t h . C A ] D ec Some formulae for products of Fubini polynomials withapplications
Levent KargınAkseki Vocational School, Alanya Alaaddin Keykubat University,Antalya TR-07630, Turkey
E-mail: [email protected]
Abstract
In this paper we evaluate sums and integrals of products of Fubini poly-nomials and have new explicit formulas for Fubini polynomials and numbers.As a consequence of these results new explicit formulas for p -Bernoulli num-bers and Apostol-Bernoulli functions are given. Besides, integrals of productsof Apostol-Bernoulli functions are derived. Key words:
Fubini numbers and polynomials, Apostol-Bernoulli functions, p -Bernoulli numbers. Let (cid:8) nk (cid:9) be the Stirling numbers of the second kind ([18]). Fubini polynomialsare defined by [29] F n ( y ) = n X k =0 (cid:26) nk (cid:27) k ! y k . (1)They have the exponential generating function11 − y ( e t −
1) = ∞ X n =0 F n ( y ) t n n ! , (2)and are related to the geometric series [5] (cid:18) y ddy (cid:19) m − y = ∞ X k =0 k m y k = 11 − y F m (cid:18) y − y (cid:19) , | y | < . Because of this relation Fubini polynomials are also called geometric polynomi-als. In addition, the following recurrence relation holds for the Fubini polyno-mials [11] F n +1 ( y ) = y ddy [ F n ( y ) + yF n ( y )] . (3)The n th Fubini number (ordered Bell number or geometric number) [10,19, 29], F n , is defined by F n (1) = F n = n X k =0 (cid:26) nk (cid:27) k !, (4)1nd counts all the possible set partitions of an n element set such that the orderof the blocks matters. Besides with this combinatorial property, these numbersare seen in the evaluation of the following series ∞ X k =0 k n k = 2 F n . (5)In the literature numerous identities concerned with these polynomials andnumbers were obtained [5, 6, 7, 8, 12, 16] and their generalizations are given[5, 13, 14, 17]. The main purpose of this paper is to generalize the binomialformulas [11] n X k =0 (cid:18) nk (cid:19) F k = 2 F n , n > , (6)2 n X k =0 (cid:18) nk (cid:19) ( − k F k = ( − n F n + 1 , n ≥ , (7)and the integral representation [22] Z − F n ( y ) dy = B n , n > . (8)Here B n is the Bernoulli numbers defined by the explicit formula B n = n X k =0 (cid:26) nk (cid:27) ( − k k ! k + 1 . (9)As applications of these generalizations we obtain explicit formulas for Apostol-Bernoulli functions and p -Bernoulli numbers and integrals of products of Apostol-Bernoulli functions. We use generating function technique in the proofs.Now we state our results. In this section we define two variable Fubini polynomials and obtain some basicproperties which give us new formulas for F n ( y ) . Moreover we shall considerthe sums of products of two Fubini polynomials. The sums of products ofvarious polynomials and numbers with or without binomial coefficients havebeen studied (e.g., [2, 21, 24, 25, 28, 30]).Two variable Fubini polynomials are defined by means of the following gen-erating function ∞ X n =0 F n ( x ; y ) t n n ! = e xt − y ( e t − . (10)For some special cases of (10), we have F n (0; y ) = F n ( y ) and F n (0; 1) = F n . (11)2e can rewrite (10) as ∞ X n =0 F n ( x ; y ) t n n ! = 11 − y ( e t − e xt = ∞ X n =0 F n ( y ) t n n ! ∞ X n =0 x n t n n != ∞ X n =0 " n X k =0 (cid:18) nk (cid:19) F k ( y ) x n − k t n n ! . Comparing the coefficients of t n n ! yields F n ( x ; y ) = n X k =0 (cid:18) nk (cid:19) F k ( y ) x n − k . (12)From (10) we have ∞ X n =0 [ F n ( x + 1; y ) − F n ( x ; y )] t n n ! = e xt ( e t − − y ( e t − y (cid:20) e xt − y ( e t − − e xt (cid:21) = 1 y ∞ X n =0 [ F n ( x ; y ) − x n ] t n n ! . Comparing the coefficients of t n n ! gives yF n ( x + 1; y ) = (1 + y ) F n ( x ; y ) − x n . (13)Thus, setting x = 0 and x = − yF n (1; y ) = (1 + y ) F n ( y ) , n > , (14)(1 + y ) F n ( − y ) = yF n ( y ) + ( − n , n ≥ , (15)respectively. Combining these relations with (12) gives the equations (6) and(7) which were obtained by using Euler-Siedel matrix method in [11].Now, we want to give the generalization of the binomial formula (6). Deriva-tive of (10) can be written as ∂∂t (cid:18) e xt − y ( e t − (cid:19) = xe xt − y ( e t −
1) + ye t − y ( e t − e xt − y ( e t − . Taking x = x + x − ∂∂t (cid:18) e xt − y ( e t − (cid:19) = ∞ X n =0 F n +1 ( x + x − y ) t n n ! ,xe xt − y ( e t −
1) = ( x + x − ∞ X n =0 F n ( x + x − y ) t n n !3nd ye t − y ( e t − e xt − y ( e t −
1) = y ∞ X n =0 F n ( x ; y ) t n n ! ! ∞ X n =0 F n ( x ; y ) t n n ! ! = y ∞ X n =0 n X k =0 (cid:18) nk (cid:19) F k ( x ; y ) F n − k ( x ; y ) t n n ! . By equating the coefficients of t n n ! , we get y n X k =0 (cid:18) nk (cid:19) F k ( x ; y ) F n − k ( x ; y ) = F n +1 ( x + x − y ) − ( x + x − F n ( x + x − y ) . For x = x = 0 in the above equation, using (15) give the sums of products ofthe Fubini polynomials. Theorem 1
For n ≥ , ( y + 1) n X k =0 (cid:18) nk (cid:19) F k ( y ) F n − k ( y ) = F n +1 ( y ) + F n ( y ) . (16) When y = 1 this becomes n X k =0 (cid:18) nk (cid:19) F k F n − k = F n +1 + F n . (17)Now, we investigate the sums of products of the Fubini polynomials fordifferent y values in the following theorem. Theorem 2
For n ≥ and y = y , n X k =0 (cid:18) nk (cid:19) F k ( y ) F n − k ( y ) = y F n ( y ) − y F n ( y ) y − y . (18) Proof.
The products of (10) can be written as e x t (1 − y ( e t − e x t (1 − y ( e t − y y − y e ( x + x ) t − y ( e t − − y y − y e ( x + x ) t − y ( e t − . Using the same method as in the proof of Theorem 1 we have n X k =0 (cid:18) nk (cid:19) F k ( x ; y ) F n − k ( x ; y ) = y F n ( x + x ; y ) − y F n ( x + x ; y ) y − y . x = x = 0 in the above equation gives the desired equation.As we know, for y = 1 Fubini polynomials reduce to Fubini numbers. Wenow point out (see (23)) that Fubini numbers arise for other value of y , too. Ifwe take y − y in (10) we have F n ( x ; y −
1) = ( − n F n (1 − x ; − y ) . (20)Setting x = 0 in the above equation and using the relation (14) we have thereflection formula F n ( y ) = ( − n yy + 1 F n ( − y − , n > . (21)Therefore, using (1) gives a new explicit formula for Fubini polynomials in thefollowing theorem. Theorem 3
For n > we obtain F n ( y ) = y n X k =1 (cid:26) nk (cid:27) ( − n + k k ! ( y + 1) k − . (22)Note that, when y = 1 , (22) reduce to [20, Thereom 4.2]. Moreover, from(21) we get two conclusion as F k (cid:18) − (cid:19) = 0 and F n ( −
2) = ( − n F n . (23)Thus, if we take y = − y = 1 in (18) and use the second part of (23),we obtain the alternating sums of products of Fubini numbers. Corollary 4
For n > , we have n X k =0 (cid:18) nk (cid:19) ( − k F k F n − k = (cid:26) n is odd F n ; n is even . (24)Finally, we obtain a new explicit formula for Fubini polynomials and numbersin the following theorem. Theorem 5
For y = − ,F n ( y ) = n X k =0 (cid:26) nk (cid:27) k ! y k h n +1 ( y + 1) y k + ( − k +1 i (2 y + 1) k +1 . (25) When y = 1 this becomes F n = n X k =0 (cid:26) nk (cid:27) k ! h n +2 + ( − k +1 i k +1 , n ≥ . (26) When y = − this becomes F n = n X k =0 ( − n − k (cid:26) nk (cid:27) k ! 2 k − (cid:2) n + k +1 + 1 (cid:3) k +1 n > . (27)5 roof. If we take y − in place of y in (2) we arrive at11 − y − ( e t −
1) = y − y (cid:20) y − e t + 1 y + e t (cid:21) . (28)Each of the function in the above equation can be written as11 − y − ( e t −
1) = ∞ X n =0 n F n (cid:18) y − (cid:19) t n n ! , (29)1 y − e t = yy − ∞ X n =0 F n (cid:18) y − (cid:19) t n n ! , (30)1 y + e t = yy + 1 ∞ X n =0 F n (cid:18) − y + 1 (cid:19) t n n ! . (31)By equating the coefficients of t n n ! , we have F n ( y ) = 2 n +1 (1 + y ) F n (cid:18) y y (cid:19) − (1 + 2 y ) F n ( − y ) . (32)Finally, using (1) in the right hand side of the above equation yields (25). The integrals of products of various polynomials and functions have been studied(e.g., [3, 6, 9, 23, 26]). In this section we deal with an integral for a product oftwo Fubini polynomials. First we need the following Lemma 6 and Lemma 7.
Lemma 6
For all k ≥ and n ≥ we have Z − y k F n ( y ) dy = ( − k k ! k X j =0 (cid:20) k + 1 j + 1 (cid:21) B n + j , (33) where (cid:2) nk (cid:3) is the Stirling numbers of the fist kind ([18]). Proof.
We prove (33) by induction on k. The case k = 0 of (33) is known from(8). If we integrate both sides of (3) with respect to y from − Z − F n +1 ( y ) dy = Z − y ddy [ F n ( y ) + yF n ( y )]= [ y ( F n ( y ) + yF n ( y ))] − − Z − [ F n ( y ) + yF n ( y )] dy.
6o using (8) yields the case k = 1 of (33) as Z − yF n ( y ) dy = − ( B n +1 + B n ) . Multiplying both sides of (3) with y and integrating it with respect to y from − Z − yF n +1 ( y ) dy = Z − y ddy [ F n ( y ) + yF n ( y )] . Applying integration by parts and using (8) yields the case k = 2 of (33) as2 Z − y F n ( y ) dy = B n +2 + 3 B n +1 + 2 B n . If we multiply both sides of (3) with y k and integrating it with respect to y from − Z − y k F n +1 ( y ) dy = Z − y k +1 ddy [ F n ( y ) + yF n ( y )] . Applying integration by parts to the right hand side of the above equation andconsidering Z − y k F n ( y ) dy = ( − k k ! k X j =0 (cid:20) k + 1 j + 1 (cid:21) B n + j , we have Z − y k +1 F n +1 ( y ) dy = ( − k +1 ( k + 1)! k X j =0 (cid:20) k + 1 j + 1 (cid:21) B n + j +1 + ( − k +1 ( k + 1)! k X j =0 ( k + 1) (cid:20) k + 1 j + 1 (cid:21) B n + j . Finally, the well known relations (cid:20) n + 1 k (cid:21) = n (cid:20) nk (cid:21) + (cid:20) nk − (cid:21) and (cid:20) n (cid:21) = ( n − , give that the statement is true for k + 1 . emma 7 For any non-negative integer m and j , m X k = j (cid:26) mk (cid:27)(cid:20) k + 1 j + 1 (cid:21) ( − k = ( − m (cid:18) mj (cid:19) . (34) Proof.
We rewrite this equation into matrix form by using the matrices( S ) i,j = ( − i + j (cid:20) i + 1 j + 1 (cid:21) , ( S ) i,j = (cid:26) ij (cid:27) , ( B ) i,j = (cid:18) ij (cid:19) . These can be considered as infinite matrices so that the statement we are goingto prove takes the form S S = B − , as the elementwise inverse of the matrix B is ( B ) − i,k = ( − i + k (cid:0) ik (cid:1) . The aboveequation is equivalent to S = B − S − = ( S B ) − . The matrix on the right hand side is easily decipherable. Elementwise it is(( S B ) − ) i,j = i X k =0 (cid:26) ik (cid:27)(cid:18) kj (cid:19) . The latter sum simply equals to i X k =0 (cid:26) ik (cid:27)(cid:18) kj (cid:19) = (cid:26) i + 1 j + 1 (cid:27) , as it is known (see [18, p. 251, formula (6.15)]). Hence our original statementequals to the matrix equation ( S ) − i,j = (cid:26) i + 1 j + 1 (cid:27) . This is nothing else but the reformulation of the fact that the second and signedfirst kind Stirling matrices are inverses of each other.Now, we are ready to give the integrals of products of Fubini polynomials.Using (1) we have Z − F m ( y ) F n ( y ) dy = Z − m X k =0 (cid:26) mk (cid:27) k ! y k F n ( y ) dy. Then, interchanging the sum and integral in the above equation and using (33)yield Z − F m ( y ) F n ( y ) dy = m X j =0 m X k = j (cid:26) mk (cid:27)(cid:20) k + 1 j + 1 (cid:21) ( − k B n + j . Finally, using Lemma 7 gives the following theorem.8 heorem 8
For all m ≥ and n ≥ we have Z − F m ( y ) F n ( y ) dy = ( − m m X j =0 (cid:18) mj (cid:19) B n + j . (35)Using the representation (1) in (35) and integrating termwise one obtains n X k =0 m X j =0 (cid:26) nk (cid:27)(cid:26) mj (cid:27) ( − k + j k ! j ! k + j + 1 = ( − m m X j =0 (cid:18) mj (cid:19) B n + j . This double sum identity extends (9).In order to give an application of Lemma 6, now we emphasize the summationin the right hand of (33). Rahmani [27] defined p -Bernoulli numbers as ∞ X n =0 B n,p t n n ! = F (cid:0) , p + 2; 1 − e t (cid:1) , where F ( a, b ; c ; z ) denotes the Gaussian hypergeometric function [1]. Thesenumbers can be written in terms Stirling numbers of the first kind p X j =0 ( − j (cid:20) pj (cid:21) B n + j = p ! p + 1 B n,p , n, p ≥ . From the above equation, we have p X j =0 ( − j +1 (cid:20) p + 1 j + 1 (cid:21) B n + j = ( p + 1)! p + 2 B n − ,p +1 , n ≥ , p ≥ . (36)Moreover when n is odd or even we have( − j +1 B n + j = B n + j or ( − j +1 B n + j = − B n + j , n > , respectively. Therefore we have p X j =0 (cid:20) p + 1 j + 1 (cid:21) B n + j = ( ( p +1)! p +2 B n − ,p +1 , n is odd − ( p +1)! p +2 B n − ,p +1 , n is even . Using the above equation, (33) can be written as Z − y p F n ( y ) dy = ( ( − p p +1 p +2 B n − ,p +1 , n is odd( − p +1 p +1 p +2 B n − ,p +1 , n is even , (37)where n > , p ≥ . On the other hand, using (1) in the left part of (37), a newexplicit formula for p -Bernoulli numbers is obtained.9 heorem 9 For n > and p > , B n − ,p = p + 1 p n − X k =0 (cid:26) n − k + 1 (cid:27) ( − k +1 ( k + 1)! k + p + 1 and B n,p = p + 1 p n X k =0 (cid:26) n + 1 k + 1 (cid:27) ( − k ( k + 1)! k + p + 1 . Apostol-Bernoulli functions B n ( λ ) have the following explicit expression B n ( λ ) = nλ − n − X k =0 (cid:26) n − k (cid:27) k ! (cid:18) λ − λ (cid:19) k , λ ∈ C \{ } . (38)Thus for λ = 1 , B ( λ ) = 0 , B ( λ ) = 1 λ − , B ( λ ) = − λ ( λ − , ... etc.The functions B n ( λ ) are rational functions in the second variable, λ . Thesefunctions were introduced by Apostol [4] in order to evaluate the Lerch tran-scendent (also Lerch zeta function) for negative integer values of s and alsowere studied and generalized recently in a number of papers, under the nameApostol-Bernoulli numbers.Comparing the (38) to (1) , Apostol-Bernoulli functions can be expressed byFubini polynomials as ([7]) B n +1 ( λ ) = n + 1 λ − F n (cid:18) λ − λ (cid:19) , λ ∈ C \{ } . (39)We can use this relation to obtain some new properties of B n ( λ ) . For ex-ample setting y = − λλ − in (22) we have B n +1 ( λ )( n + 1) = ( − n λ n X k =0 (cid:26) nk (cid:27) k ! (cid:18) λ − (cid:19) k +1 , λ = 1 , n ≥ , which was obtained in [20, Thereom 4.3]. Similarly, from Thereom 1 we get thesums of products of Apostol-Bernoulli functions as given in [15, Corollary 1.3]by different method. Moreover, using the equation (25) of Theorem 5 gives anew explicit formula for Apostol-Bernoulli functions. Corollary 10
For λ = ± and n ≥ , B n +1 ( λ )( n + 1) = n X k =0 (cid:26) nk (cid:27) k ! ( − λ ) k h n +1 λ k + ( λ − k +1 i ( λ − k +1 .
10o give a different application of the relation (39), first we deal with Lemma6. Replacing y with λ − λ in (33), we have Z −∞ λ k ( λ − k +1 B n +1 ( λ ) dλ = n + 1 k ! k X j =0 (cid:20) k + 1 j + 1 (cid:21) B n + j , where k ≥ n ≥ . Similarly, from Theorem 8 we obtain the integrals ofproducts of Apostol-Bernoulli functions as given in the following corollary.
Corollary 11
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