aa r X i v : . [ m a t h . N T ] N ov On q -deformed Stirling numbersYilmaz Simsek University of Akdeniz, Faculty of Arts and Science, Department of Mathematics, 07058Antalya , Turkey
E-Mail: [email protected],Abstract
The purpose of this article is to introduce q -deformed Stirling numbers of the first and second kinds.Relations between these numbers, Riemann zeta function and q -Bernoulli numbers of higher order are given.Some relations related to the classical Stirling numbers and Bernoulli numbers of higher order are found.By using derivative operator to the generating function of the q -deformed Stirling numbers of the secondkinds, a new function is defined which interpolates the q -deformed Stirling numbers of the second kinds atnegative integers. The recurrence relations of the Stirling numbers of the first and second kind are given. Inaddition, relation between q -deformed Stirling numbers and q -Bell numbers is obtained. Key Words and Phrases. q -Bernoulli numbers and polynomials, q -Stirling numbers first and secondkind, fermionic Stirling numbers first and second kind. q -Bell Numbers.1. Introduction, Definitions and Notations
The q -deformed Stirling numbers of the first and second kind are denoted by s ( n, k, q ), S ( n, k, q ), re-spectively. The fermionic Stirling numbers of the first and second kind are denoted by s f ( n, k ), S f ( n, k ),respectively. In this paper, we use notation in the work of Kim[9] and Schork[20]. q -Stirling numbers werefirst defined in the work of Carlitz[1]. A lot of combinatorial work has centered around the q -analogue, theearliest by Milne[17]; also see ([20], [4], [6], [9], [15], [16], [22]). In [8], Kim constructed q -Bernoulli numbersof higher order associated with the p -adic q -integers. q -Volkenborn integral is originately constructed byKim[9]. By using the q -Volkenborn integral, Kim[9] evaluated complete sum for q -Bernoulli polynomials. Healso obtained relations between q -Bernoulli numbers and q -analogs of the Stirling numbers. The fermionicand bosonic Stirling numbers were given in detail by Schork[20], [21]. In [22], Wagner studied three partitionstatistics and the q -Stirling and q -Bell numbers that serve as their generating functions, evaluating thesenumbers when q = − q -deformed Stirling numbers of the second kinds. Relations between these numbers,Riemann zeta functions and q -Bernoulli numbers higher order are given. We also give some relations relatedto the classical Stirling numbers and Bernoulli numbers higher order. By using derivative operator tothe generating function of the q -deformed Stirling numbers of the second kinds, we define new function,which interpolates the q -deformed Stirling numbers of the second kinds at negative integers. The recurrencerelations of the Stirling numbers of the first and the second kind are given. We also give relation between q -deformed Stirling numbers and q -Bell numbers.Let q ∈ ( − , q -integers and q -deformed numbers are given by[ n, q ] = [ n ] = 1 − q n − q . (1.1) [ n ]! = [ n ][ n − ... [2][1], [0]! = 1 and (cid:18) nk (cid:19) q = [ n ]![ n − k ]![ k ]! . Note that lim q → [ n ] = n , cf. ([6], [7], [10], [11], [12], [20], [13], [14], [15]).The generating function of the q -Stirling numbers of the second kind is given by defined by[9]: F S,q ( t ) = q k (1 − k )2 [ k ]! k X j =0 ( − k − j (cid:18) kj (cid:19) q q ( k − j )( k − j − e [ j ] t = ∞ X n =0 S ( n, k, q ) t n n ! . (1.2)Let ( Eh )( x ) = h ( x + 1) be the shift operator. Let ∆ nq = Q n − j =0 ( E − q j I ) be the q -difference operator cf. [9].[ x ] n = n X k =0 (cid:18) xk (cid:19) q [ k ]! q k (1 − k )2 S ( n, k, q ) cf. ([9]),where S ( n, k, q ) is denoted the q -Stirling numbers of the second kind. By the above equation, we have S ( n, k, q ) = q k (1 − k )2 k ]! k X j =0 ( − j q j ( j − (cid:18) kj (cid:19) q [ k − j ] n cf. ([9]).Kim[9] defined q -analog of the Newton-Gregory expansion as follows: S ( n, k, q ) = q k (1 − k )2 [ k ]! ∆ kq n q -deformed Stirling Numbers When q <
0, we write q ≡ − q ∼ with q ∼ >
0. By (1.1), we have[ n ] = 1 − ( − q ∼ ) n q ∼ = [ n, q ∼ ] F cf. ([20]),[ n, q ∼ ] F is called q ∼ -fermionic basic numbers. These numbers are appearing in recent studies of the q ∼ -deformed fermionic oscillator (see [16], [20], [18], [21]). Observe that q → −
1, i. e., q ∼ → n, q = −
1] = [ n, q ∼ = 1] F = 1 − ( − n ǫ n = (cid:26)
0, if n is even integers1, if n is odd integers. (2.1)Hence, for n ≥ n, q = − . (2.2)The q ∼ -fermionic basic numbers were given in detail by Schork[20], [21].By using (1.2), the q -deformed Stirling numbers of the second kind, S ( n, k, q ) are given by (in the versionof Kim[9]): S ( n, k, q ) = k X j =0 ( − k − j q k (1 − k )+( k − j )( k − j − [ j ] n − [ j − k − j ]! . (2.3)where n, k ∈ N with k ≤ n . The recurrence relations of S ( n, k, q ), with S (1 , , q ) = 0 and S (1 , , q ) = 1, isgiven by S ( n + 1 , k, q ) = q k − S ( n, k − , q ) + [ k ] S ( n, k, q ), cf. ([2],[4],[9],[17], [20]).The recurrence relation implies that S ( n, k, q ) are polynomials in q (for detail see [20]). In the bosonic limit: lim q → S ( n, k, q ) = S ( n, k ). Some special values of k , the q -deformed Stirling numbers of the secondkind S ( n, k, q ) are given by[20]: S ( n, , q ) = 1, S ( n, , q ) = [2] n − − S ( n, n, q ) = q n ( n − .By applying the derivative operator d k dt k F S,q ( t ) | t =0 to (1.2), we arrive at the following theorem: S ( n, k, q ) = d n dt n F S,q ( t ) | t =0 = q k (1 − k )2 [ k ]! k X j =0 ( − k − j (cid:18) kj (cid:19) q q ( k − j )( k − j − [ j ] n . (2.4)By using (2.4), we define Y S ( z, k, q ) function as follows: Definition 1.
Let z ∈ C . We define Y S ( z, k, q ) = q k (1 − k )2 [ k ]! k X j =0 ( − k − j (cid:18) kj (cid:19) q q ( k − j )( k − j − [ j ] − z . (2.5)Observe that if z ∈ C , then Y S ( z, k, q ) is an analytic function. The function Y S ( z, k, q ) interpolates the q -deformed Stirling numbers of the second kind S ( n, k, q ) at negative integers, which is given in Theorem 1,below.By (1.1), (2.4), we have the following Corollary: Corollary 1. S ( n, k, q ) = q k (1 − k )2 (1 − q ) n [ k ]! k X j =0 n X d =0 ( − k − j − d (cid:18) kj (cid:19) q (cid:18) nd (cid:19) q ( k − j )( k − j − jd . By substituting z = − n , with n is a positive integer, into (2.5), and using (2.4), we arrive at the followingtheorem: Theorem 1.
Let n be a positive integer. Then we have Y S ( − n, k, q ) = S ( n, k, q ) . (2.6)The q -deformed Bell numbers are defined by [20] B ( n, q ) = n X k =0 S ( n, k ; q ) . By using (2.4) and (2.6), we give relation between Y S ( z, k, q ) and B ( n, q ) as follows: Theorem 2.
Let n be a positive integer. Then we have B ( n, q ) = n X k =0 Y S ( − n, k, q )= n X k =0 k X j =0 ( − k − j (cid:18) kj (cid:19) q [ j ] n q k (1 − k )+( k − j )( k − j − [ k ]! . Remark 1.
Observe that when lim q → B ( n, q ) = B ( n ) = P nk =0 S ( n, k ) , where B ( n ) denotes the classicalBell numbers cf. ( [20] , [17] , [22] ). In [5] , Gessel gave relation between the classical Stirling numbers of firstkind, S ( n, k ) and the classical Bernoulli numbers of higher order, B ( n ) k as follows: S ( n + k, n ) = (cid:18) n + kk (cid:19) B ( − n ) k , (2.7) where ∞ X j =0 B ( n ) j t j j ! = (cid:18) te t − (cid:19) n . In [9] , Kim gave relation between S ( n, k ; q ) numbers and q -Bernoulli numbers of higher-order as follows: Let m ≥ and h, k are natural numbers. m k m X j =0 (cid:18) mj (cid:19) ( q − j β j ( o, − k, q ) = k X j =0 q k ( k − [ j ]! S ( k, j ; q ) (cid:18) mj (cid:19) q , where β m ( h, k, q ) = (1 − q ) − m m X j =0 (cid:18) mj (cid:19) ( − j (cid:18) h + j [ h + j ] (cid:19) k . In [19] , Rassias and Srivastava gave relation between Riemann zeta functions and the classical Stirlingnumbers of first kind, s ( n, k ) as follows: ζ ( k + 1) = ∞ X n = k ( − n − k n.n ! s ( n, k ) . (2.8)For each k = 0 , , ..., n −
1, ( n ≥ Eulerian numbers E ( n, k ) are given by[3] E ( n, k ) = k X j =0 ( − j (cid:18) n + 1 j (cid:19) ( k + 1 − j ) n . Relation between E ( n, k ) and the classical Stirling numbers of the second kind, S ( n, k ) is given by[3] S ( n, m ) = 1 m ! n − X j =0 E ( n, j ) (cid:18) jn − m (cid:19) , n ≥ m, n ≥ . (2.9)By (2.7) and (2.9), after some elementary calculations, we arrive at the following corollary: Corollary 2.
Let k = 0 , , ..., n − , ( n ≥ ). Then we have B ( − n ) k = (cid:18) n + kk (cid:19) n ! n + k − X j =0 E ( n + k, j ) (cid:18) jk (cid:19) . (2.10)By (2.6) with q → Corollary 3.
Let n be a positive integer and n ≥ k, n ≥ . Then we have Y S ( − n, k ) = 1 k ! n − X j =0 E ( n, j ) (cid:18) jn − k (cid:19) . The q -deformed Stirling number of the first kind s ( n, k, q ), with s (1 , , q ) = 0 and s (1 , , q ) = 1, satisfythe following recurrence relations ( see [20], [2]): s ( n + 1 , k, q ) = q − n ( s ( n, k − , q ) − [ n ] s ( n, k, q )) . (2.11)If q → s ( n + 1 , k ) = s ( n, k − − ns ( n, k )For k ≥
1, we set [ x ] k = [ x ][ x − ... [ x − k + 1] , cf. ([1], [6], [20], [21], [18]).By using the above relation, the q -deformed Stirling numbers is defined as [20][ x ] n = n X j =0 S ( n, j, q )[ x ] j and [ x ] n = n X j =0 s ( n, j, q )[ x ] j . (2.12)By using (2.12), the q -deformed Stirling numbers of the first and the second kind satisfies for n ≥ m theinversion relations, which are given by the following theorem: Theorem 3.
Let n and m be non-negative integers. Then we have n X k = m s ( n, k, q ) S ( k, m, q ) = δ n,m , n X k = m S ( n, k, q ) s ( k, m, q ) = δ n,m . Proofs of this theorem were given by Schork[20] and Charalambides[2]. From the above theorem, q -Stirlingnumbers of the first and the second kind satisfy the ortogonality relations. Further Remarks and Observations
The fermionic Stirling numbers of the first and the second kind studied by [17], [20], [21], [1], [4], [22],[15], [16], [18], [6], [9]. In this section, we can use some notations which are due to Schork[20], and Kim[9].In [20], Schork gave the recurrence relations of the fermionic Stirling numbers of first kind and second kindas follows: s f ( n + 1 , k ) = ( − n s f ( n, k −
1) + ( − n +1 ǫ n s f ( n, k ) , (3.1)with s f (1 ,
0) = 0 and s f (1 ,
1) = 1. For the convention, here we take s f ( n,
0) = 0 and S f ( n + 1 , k ) = ( − k − S f ( n, k −
1) + ǫ k S f ( n, k ) , (3.2)with S f (1 ,
0) = 0 and S f (1 ,
1) = 1, where ǫ k is defined in (2.1). Schork gave the values S f ( n, k ) for maximaland small k and found S f ( n, n ) = ( − n ( n − as well as S f ( n,
1) = 1, S f ( n,
2) = − S f ( n,
3) = 2 − n , S f ( n,
4) = n − s f ( n, n ) = ( − n ( n − .By (3.1) and (2.1), many Stirling numbers of the first kind vanish. By induction over k , Schork[20] prove s f ( n, k ) = 0 for n > k . By the same method, s f ( n, k ∼ ) = 0 for 1 ≤ k ∼ ≤ k and n > k ∼ and s f ( n, k +1) = 0for n ≥ k + 3 (for detail see [20]). From (3.1), we arrive at the following theorem[20]: Theorem 4.
Let k be non-negative integer. Then we have s f (2 k + 3 , k + 1) = s f (2 k + 2 , k ) − ǫ k +2 s f (2 k + 2 , k + 1) . Note that by the induction hypothesis, the first summand vanishes, whereas by the ǫ k +2 = 0, the secondsummand vanish. Hence, for a given n , the first n Stirling numbers s f ( n, k ) vanish. Let[ n ] f = [ n, q = −
1] = ǫ n cf. [20]for the fermionic basic numbers, by (2.1). The fermionic Stirling numbers are connection coefficients for thefermionic basic numbers [20]:[ x ] nf = n X j =0 S f ( n, j )[ x ] jf and [ x ] nf = n X j =0 s f ( n, j )[ x ] jf . (3.3)By induction over n , Schork[20] proved the first equation. We give sketch of the proof as follows: Let theassertion obtain for n . Thus, the induction hypothesis implies that[ x ] n +1 f = n X j =0 S f ( n, j )[ x ] jf [ x ] f . By using f -arithmetic relations, [ x ] f = [ j ] f +( − j [ x − j ] f , [ n + m ] f = [ m ] f +( − m [ n ] f and [ n +1] f = 1 − [ n ] f ,we easily find [ x ] n +1 f = n X j =0 S f ( n, j )[ x ] jf [ x ] f = n X j =0 (cid:0) S f ( n, j )[ j ] f + ( − j [ x − k ] f S f ( n, j ) (cid:1) [ x ] jf = n X j =0 S f ( n, j )[ j ] f [ x ] jf + n X j =0 ( − j [ x − k ] f S f ( n, j )[ x ] jf = n X j =0 S f ( n, j )[ j ] f [ x ] jf + n X j =0 ( − j S f ( n, j )[ x ] j +1 f = n +1 X j =1 (cid:0) S f ( n, j )[ j ] f + ( − j − S f ( n, j − (cid:1) [ x ] jf . (3.4) Since S f ( n + 1 ,
0) = 0, we have[ x ] n +1 f = n +1 X j =0 S f ( n + 1 , j )[ x ] jf = n +1 X j =1 S f ( n + 1 , j )[ x ] jf . (3.5)By (3.4) and (3.5), we arrive at (3.2).Since [ x − n ] f = ( − n [ x ] f + ( − n +1 ǫ n , we have[ x ] n +1 f = [ x − n ] f n X j =0 s f ( n, j )[ x ] jf = n X j =0 s f ( n, j ) (cid:0) ( − n [ x ] f + ( − n +1 ǫ n (cid:1) [ x ] jf = n X j =0 ( − n s f ( n, j )[ x ] j +1 f + n X j =0 ( − n +1 s f ( n, j ) ǫ n [ x ] jf = n +1 X j =1 (cid:0) ( − n s f ( n, j −
1) + ( − n +1 s f ( n, j ) ǫ n (cid:1) [ x ] jf , by the above equation, we obtain (3.1).Observe that by (3.1) and (3.2), Schork[20] defined the fermionic Stirling numbers of first and secondkind, respectively.It is well-known that if j ≥
2, then [ x ] jf vanishes, so by using the first equation of (3.3)[ x ] nf = S f ( n, x ] f = [ x ] f . It is easy to prove the following theorem.
Theorem 5.
Let n and m be non-negative integers. Then we have n X j = m s f ( n, j ) S f ( j, m ) = δ n,m , and n X j = m S f ( n, j ) s f ( j, m ) = δ n,m . (3.6)By (3.6), the fermionic Stirling numbers satisfy the inversion relations. The proof of (3.6), was given bySchork[20]. He also proved the following the recurrence relations of the fermionic Stirling numbers of secondkind: For odd j > S f ( n + 1 , j ) = S f ( n, j ) + S f ( n, j − n and j be non-negativeintegers. S f ( n + 1 , j ) = S f ( n, j ) − S f ( n − , j − . Acknowledgement 1.
We would like to thank to Professor Matthias Schork for his many valuable sug-gestions and comments on this paper. We also would like to thank to Professor Taekyun Kim for his manyvaluable comments on this paper.This work was supported by Akdeniz University Scientific Research Projects Unit.
References [1] L. Carlitz, q -Bernoulli numbers and polynomials, Duke Math. 15 (1948) 987-1000.[2] Charalambides, C. A., Non-central generalized q -factorial coefficients and q -Stirling numbers, Discrete Math., 275 (2004)67-85.[3] G-S. Cheon and J-S. Kim, Factorial Stirling matrix and related combinatorial sequences, Linear algebra Appl., 357 (2002),247-258.[4] R. Ehrenborg, Determinants involving q -Stirling numbers, Advan. Appl. Math. 31 (2003) 630-642.[5] Gessel, I. M., On Miki’s identity for Bernoulli numbers, J. Number Theory, 110 (2005) 75-82.[6] J. Katriel, Stirling numbers Identity: Interconsistency of q -analogues, J. Phys. A 31(15) (1998) 3559-3672.[7] T. Kim, On a q -analogue of the p -adic log gamma functions and related integrals, J. Number Theory 76(2) (1999) 320–329.[8] T. Kim, Some q -Bernoulli numbers of higher order associated with the p -adic q -integers. Proc. Jangjeon Math. Soc. 2(2001), 23-28. R03.11 - 13A.323[9] T. Kim, q -Volkenborn integration, Russ. J. Math Phys. 19 (2002) 288-299.[10] T. Kim, On p -adic q - L -functions and sums of powers, Discrete Math. 252(1-3) (2002) 179–187. [11] T. Kim, An invariant p -adic integral associated with Daehee numbers, Integral Transforms Spec. Funct. 13(1) (2002) 65-69.[12] T. Kim, Non-Archimedean q -integrals associated with multiple Changhee q -Bernoulli polynomials, Russian J. Math. Phys.10 (2003) 91-98.[13] T. Kim, Sums powers of consecutive q-integers, Advan. Stud. Contemp. Math. 9 (2004) 15-18.[14] T. Kim, S. D. Kim and D. -W. Park, On uniform differentiability and q -Mahler expansions, Adv. Stud. Contep. Math.4(1) (2001) 35-41.[15] A. K. Kwasniewski, q -Poisson, q -Dobinski, q -Rota and q -coherent states-a second fortieth anniversary memoir, Proc.Jangjeon Math. Soc. 7(2) (2004) 95–98.[16] A. K. Kwasniewski, On psi-umbral extension of Stirling numbers and Dobinski-like formulas, Advan. Stud. Contemp. Math.12 (2006) 73-100.[17] S. C. Milne, A q-analog of restricted growth functions, Dobinski’s equality, and Charlier polynomials, Trans. Amer. Math.Soc. 245 (1978) 89-118.[18] Parthasarathy R., q -fermionic numbers and their roles in some physical problems, Phys. Lett. A, 326 (2004) 178-186.[19] T. M. Rassias and H. M. Srivastava, Some classes of infinite series associated with the Riemann zeta and polygammafunctions and generalized harmonic numbers, Appl. Math. Comput. 131 (2002) 593-605.[20] M. Schork, Fermionic relatives of Stirling and Lah numbers, J. Phys. A: Math. Gen. 36 (2003) 10391-10398.[21] M. Schork, Normal ordering q -bosons and combinatorics, Phys. Lett. A, 335(2006), 293-297.[22] C. G. Wager, Partition statistics and q -Bell numbers ( q = −−