Applied Mathematics and Computation 219 (2013) 9978–9991
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Applied Mathematics and Computation
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The generalized Touchard polynomials revisited ⇑ Toufik Mansour a, , Matthias Schork b a Department of Mathematics, University of Haifa, 31905 Haifa, Israel b Camillo-Sitte-Weg 25, 60488 Frankfurt, Germany article info abstract
Keywords: We discuss the generalized Touchard polynomials introduced recently by Dattoli et al. as Touchard polynomial well as their extension to negative order introduced by the authors with operational meth- Stirling number ods. The connection to generalized Stirling and Bell numbers is elucidated and analogs to Bell number Burchnall’s identity are derived. A recursion relation for the generalized Touchard polyno- Generating function mials is established and it is shown that one can interpret some of the resulting formulas as binomial theorems for particular noncommuting variables. We suggest to generalize the generalized Touchard polynomials still further and introduce so called Comtet–Touchard functions which are associated to the powers of an arbitrary derivation. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction
The Touchard polynomials (also called exponential polynomials) may be defined (see, e.g., [1,2,6,16,17]) for n 2 N by d n T ðxÞ :¼ e x x ex: ð1:1Þ n dx
d In the following we also use the notation D ¼ @x ¼ dx. If we further denote the operator of multiplication with x by X, i.e., ðXf ÞðxÞ¼xf ðxÞ for allP functions f considered, we can make contact with operational formulas [16,17]. For example, using n n k k the fact that ðXDÞ ¼ k¼0Sðn; kÞX D , where Sðn; kÞ denotes the Stirling numbers of the second kind, one obtains directly from the definition of the Touchard polynomials the relation
Xn k TnðxÞ¼ Sðn; kÞx ¼ BnðxÞ; ð1:2Þ k¼0 where the second equality corresponds to the definition of the conventional Bell polynomials. Dattoli et al. introduced in [6, Eq. (36)] Touchard polynomials of higher order. They are defined for m 2 N (and n 2 N)by
ðmÞ x m n x Tn ðxÞ :¼ e ðx @xÞ e ; ð1:3Þ and reduce for m ¼ 1 to the conventional Touchard polynomials from above. Many of their properties are discussed in [6].In m n particular, noting that the normal ordering of ðx @xÞ leads to the generalized Stirling numbers Sm;1ðn; kÞ considered, e.g., by
Lang [12], one has a close connection between the Touchard polynomials of order m and the Stirling numbers Sm;1ðn; kÞ.It was mentioned in [6, Eq. (37)] that the higher order Touchard polynomials satisfy the recursion relation
m m ðmÞ ðmÞ ðx þ x @xÞTn ðxÞ¼Tnþ1ðxÞ: ð1:4Þ
⇑ Corresponding author. E-mail addresses: toufi[email protected], [email protected] (T. Mansour), [email protected] (M. Schork).
0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.04.010 T. Mansour, M. Schork / Applied Mathematics and Computation 219 (2013) 9978–9991 9979
In [15] the Touchard polynomials of negative integer order m (with m 2 N) are defined for all n 2 N by
ð mÞ x m n x Tn ðxÞ :¼ e ðx @xÞ e ; ð1:5Þ and several of their properties are discussed (in close analogy to the Touchard polynomials of higher order considered in [6]). For example, from the definition above, it is easy to see that the analog of (1.4) holds true, i.e.,
m m ð mÞ ð mÞ ðx þ x @xÞTn ðxÞ¼Tnþ1 ðxÞ: ð1:6Þ
ð0Þ x n x Note that one can also define the Touchard polynomials of order zero, but due to Tn ðxÞ¼e @x e ¼ 1 for all n 2 N,no interesting polynomials result. As a particular example let us mention that one has a very close connection (mentioned ð 1Þ in the main text below) between Tn ðxÞ and Bessel polynomials. In the general case it was shown in [15] that the Tou- chard polynomials (of arbitrary order m 2 ZÞ have a nice expression in terms of the generalized Stirling and Bell num- bers discussed by the authors in [13,14]. The aim of the present paper is to continue the study of the Touchard polynomials of negative order. Several results will be derived in analogy to the case of Touchard polynomials of higher order (treated in [6]) and new results will be given for arbitrary order, e.g., a new recursion relation. Furthermore, sev- eral connections of the generalized Touchard polynomials to other combinatorial objects will be drawn and a further generalization is suggested. Let us outline the structure of the paper in more detail. In Section 2 we consider the Touchard polynomials of order 1 in detail since it is possible to derive all results completely explicit. In particular, using these results it is possible to find an explicit binomial formula for variables U; V satisfying UV ¼ VU V 3 involving Bessel polynomials. In Section 3 the Touchard polynomials of arbitrary negative integer order are discussed along the same lines. A recursion relation for the generalized Touchard polynomials of arbitrary integer order is derived in Section 4. In Section 5 it is suggested to consider generalized Touchard functions of arbitrary real order. As a particular example the case of order 1=2 is dis- cussed and it is shown that the associated Touchard functions are given by Hermite polynomials. In Section 6 we moti- vate a further generalization and introduce so called Comtet–Touchard functions associated to powers of an arbitrary derivation. Finally, in Section 7 some conclusions are presented and some possible avenues for future research are outlined.
2. The Touchard polynomials of order 1
It was observed in [15] that for the Touchard polynomials of negative order the first nontrivial case m ¼ 1 seems to be much nicer than the general case. This is the reason why we treat in the present section this case explicitly before turning to the general case in the next section. From (1.6) we obtain in the case m ¼ 1 that
1 1 ð 1Þ ð 1Þ ðx þ x @xÞTn ðxÞ¼Tnþ1 ðxÞ:
1 1 k ð 1Þ ð 1Þ Introducing the operator M1 :¼ðx þ x @xÞ we can denote this as M1Tn ðxÞ¼Tnþk ðxÞ, yielding X k t ð 1Þ 1 T ðxÞ¼etM1 Tð ÞðxÞ: ð2:1Þ k! nþk n kP0 To understand the right-hand side, we have to study the operator
1 1 etM1 ¼ etðx þx @xÞ; ð2:2Þ closer, i.e., we need a disentanglement identity which would allow us to write this as a product of two operators where each factor depends on only one of the operators. Let us recall from [8, Eq. (I.2.34)] the following crucial result: Given two oper- ators A; B satisfying ½A; B ¼mAn, one has the following disentanglement identity "# no 1 n 1 n 2 eAþB ¼ exp ð1 þ mðn 1ÞA Þn 1 1 eB: ð2:3Þ mðn 2ÞAn 2
1 1 For the case we are interested in we identify A tx and B tx @x with commutation relation ½A; B ¼t 1A3: Thus, we can use (2.3) with m ¼ t 1 and n ¼ 3 to find after some simplifications hino 1 2 1 eAþB ¼ exp tA ð1 þ 2t 1A Þ2 1 eB; which can be expressed in terms of the original operators as pffiffiffiffiffiffiffiffiffi 1 1 2 1 etðx þx @xÞ ¼ e x þ2t xetx @x : ð2:4Þ This is the sought-for disentanglement identity. Inserting this into (2.1), we obtain 9980 T. Mansour, M. Schork / Applied Mathematics and Computation 219 (2013) 9978–9991
X k pffiffiffiffiffiffiffiffiffi t 2 1 Tð 1ÞðxÞ¼e x þ2t xetx @x Tð 1ÞðxÞ: k! nþk n kP0 Recalling from [15, Corollary 6.15] that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ekx @x f ðxÞ¼f x2 þ 2k ; ð2:5Þ we have shown the following proposition as analog to [6, Eq. (38)].
Proposition 2.1. The Touchard polynomials of order 1 satisfy the relation
X k pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t 2 Tð 1ÞðxÞ¼e x þ2t xTð 1Þ x2 þ 2t : ð2:6Þ k! nþk n kP0 pffiffiffiffiffiffiffiffiffi x2þ2t x ð 1Þ According to [15, Corollary 6.15], e is the exponential generating function of the Tn ðxÞ. It follows that ()() X k pffiffiffiffiffiffiffiffiffi X m X l t 1 1 2 1 t t ðx 1 þ x 1@ Þk ¼ etðx þx @xÞ ¼ e x þ2t xetx @x ¼ Tð 1ÞðxÞ ðx 1@ Þl x k! m! m l! x kP0 mP0 lP0 () X ts Xs s ¼ Tð 1ÞðxÞðx 1@ Þr : s! s r x sP0 r¼0 r Comparing coefficients yields the following corollary as analog to [6, Eq. (39)].
Corollary 2.2. For k 2 N0 one has the operational relation
Xk 1 1 k k ð 1Þ 1 r ðx þ x @xÞ ¼ Tk r ðxÞðx @xÞ : ð2:7Þ r¼0 r ð 1Þ In the particular case we are considering one has a beautiful expression for T‘ ðxÞ in terms of Bessel polynomials. The n- th Bessel polynomial (with n 2 N0) is defined by
Xn ðn þ kÞ! : k; : ynðxÞ ¼ k x ð2 8Þ k¼0 2 k!ðn kÞ! see, e.g., [20] where also some properties of these polynomials as well as those of the corresponding Bessel numbers can be found. The first few Bessel polynomials are given by
2 y0ðxÞ¼1; y1ðxÞ¼1 þ x; y2ðxÞ¼1 þ 3x þ 3x : ð2:9Þ According to [15, Theorem 6.10], one has the relation 1 Tð 1ÞðxÞ¼x ‘y : ‘ ‘ 1 x
This is valid for ‘>0, but if we set for convenience y 1ðxÞ¼1 it also holds true for ‘ ¼ 0. Using this relation, (2.7) can be written alternatively as Xk 1 1 k k ðk rÞ 1 1 r ðx þ x @xÞ ¼ x yk r 1ð x Þðx @xÞ : ð2:10Þ r¼0 r This identity can now be interpreted in a straightforward fashion as a particular example of a noncommutative binomial the- orem.In[13] variables U; V were considered which satisfy the commutation relation UV ¼ VU þ hV s; ð2:11Þ for arbitrary s 2 R (and h – 0). In the particular case s ¼ 0 and h ¼ 1 this reduces to the commutation relation of the Weyl algebra generated by fX;@xg satisfying @x X ¼ X @x þ 1 (thus, U # @x and V # X). For variables U; V satisfying (2.11) expressions for ðU þ VÞk were discussed in [13] (see also the many references to the literature for particular choices of 1 1 parameters s; h given therein). In the case at hand we identify X # V and X @x # U and find that 1 1 1 1 1 3 X @x X ¼ X X @x ðX Þ ,or UV ¼ VU V 3: ð2:12Þ
1 1 Thus, the algebra generated by fX ; X @xg (with the above commutation relation) yields an example for (2.11) with s ¼ 3 and h ¼ 1. The identity (2.10) can then be interpreted as follows. T. Mansour, M. Schork / Applied Mathematics and Computation 219 (2013) 9978–9991 9981
3 Theorem 2.3. Let U; V be variables satisfying UV ¼ VU V . Then the following binomial formula holds true for any k 2 N0 Xk k k k r r ðV þ UÞ ¼ V yk r 1ð VÞU ; ð2:13Þ r¼0 r where y‘ are the Bessel polynomials introduced in (2.8).
In the beautiful formula (2.13) the factor yk r 1ð VÞ arises due to the noncommutative nature of the variables and com- pletely describes it (for commuting variables U; V this factor is absent in the binomial formula).
Example 2.4. Let us consider the first few instances of (2.13). For k ¼ 0 the identity (2.13) reduces to the trivial statement 1 ¼ 1 (note that we have here used the convention y 1ðxÞ¼1). For k ¼ 1 the left-hand side yields V þ U, whereas the right- 1 1 hand side yields explicitly Vy ð VÞU0 þ V0y ð VÞU1 ¼ V þ U, where we have used the convention y ð VÞ¼1 0 0 1 1 1 as well as the explicit expression y0ð VÞ¼1, see (2.9). For k ¼ 2 the right-hand side of (2.13) gives the explicit expression 2 2 2 V 2y ð VÞU0 þ Vy ð VÞU þ V 0y ð VÞU2 ¼ V 2 þ 2VU V 3 þ U2; 0 1 1 0 2 1 where we have used the convention y 1ð VÞ¼1 as well as the explicit expressions y0ð VÞ¼1 and y1ð VÞ¼1 V, see (2.9). On the other hand, computing ðV þ UÞ2 ¼ðV þ UÞðV þ UÞ¼V 2 þ VU þ UV þ U2 and using the commutation relation UV ¼ VU V 3 leads to the same result, as it should. As a final example, we consider k ¼ 3 where the right-hand side of (2.13) gives the explicit expression 3 3 3 3 V 3y ð VÞU0 þ V 2y ð VÞU þ Vy ð VÞU2 þ V 0y ð VÞU3: 0 2 1 1 2 0 3 1
Using the convention y 1ð VÞ¼1 as well as the explicit expressions given in (2.9), this equals V 3ð1 3V þ 3V 2Þþ3V 2ð1 VÞU þ 3VU2 þ U3: On the other hand, one may compute directly
ðV þ UÞ3 ¼ðV þ UÞðV þ UÞ2 ¼ðV þ UÞðV 2 V 3 þ 2VU þ U2Þ; which gives – after using several times the commutation relation – the same result. Note that the highest power of V which appears is V 5. From (2.13) one reads off that in general the highest power of V which appears in the explicit expression of ðV þ UÞk is V 2k 1.
Remark 2.5. It is interesting to compare the above result to the ones derived in [13]. For example, it was shown in [13, Prop- osition 5.2] that one has for variables U; V satisfying (2.11) that (we have switched the notation to the one used here) ! Xk kX r 1 k i ðsÞ k rþiðs 2Þ r ðV þ UÞ ¼ h dk ðr; iÞV U ; ð2:14Þ r¼0 i¼0
ðsÞ where the coefficients dk ðj; iÞ satisfy a particular recursion relation (and their generating function is stated in [13, Theo- rem 5.3]). This expression reduces for s ¼ 3! and h ¼ 1to Xk kX r 1 k k r i ð3Þ r ðV þ UÞ ¼ V ð VÞ dk ðr; iÞ U ; r¼0 i¼0 which, upon comparison with (2.13), allows us to conclude that kX r 1 k i ð3Þ yk r 1ð VÞ¼ ð VÞ dk ðr; iÞ: r i¼0
Using the explicit expression for ynðxÞ given in (2.8), one finds ð3Þ k ðk r þ i 1Þ! dk ðr; iÞ¼ : r 2ii!ðk r i 1Þ! Before we close this section, we would like to point out how the above methods can be used to derive a summation for- mula for Bessel functions in a slightly alternative fashion to [7]. Let us denote for l 2 N by JlðxÞ the cylindrical Bessel function ÀÁ k 1 d k l ð 1Þ which satisfy the identity x dx fx JlðxÞg ¼ xkþl JlþkðxÞ, see [19, Section 17–211]. If we consider l ¼ 0 as well as k ¼ n, then sn multiplication with n! on both sides and summing over n yields X n X n sn 1 d ð 1Þ sn J ðxÞ¼ J ðxÞ; n! x dx 0 xn n! n nP0 nP0 9982 T. Mansour, M. Schork / Applied Mathematics and Computation 219 (2013) 9978–9991 which is Eq. (8) of [7]. Letting s ¼ tx, one obtains X n s 1 d t e ðÞxdx J ðxÞ¼ J ðxÞ: 0 n! n nP0 ÀÁpffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Using (2.5), the left-hand side equals J0 x þ s , giving finally the identity [7, Eq. (16)]. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X tn J x2 2xt ¼ J ðxÞ: 0 n! n nP0
3. The Touchard polynomials of arbitrary negative integer order
In the previous section we considered the Touchard polynomials of order 1. In this section we transfer the above treat- m m ment to the Touchard polynomials of arbitrary negative (integer) order m. Introducing the operator Mm :¼ðx þ x @xÞ, we can write as above X tk Tð mÞðxÞ¼etMm Tð mÞðxÞ ð3:1Þ k! nþk n kP0 and have to study the operator
m m etMm ¼ etðx þx @xÞ: ð3:2Þ
m m Identifying A tx and B tx @x, one obtains the commutation relation
1 2mþ1 ½A; B ¼mt mA m : ð3:3Þ
1 Thus, we can use the disentanglement identity (2.3) (with parameters mt m and 2mþ1) to find after some simplifications m 1 A B 1 1 1 mþ1 mþ1 B e þ ¼ exp ðtA Þm 1 þðm þ 1Þt mA m 1 e :
This can be expressed in terms of the original operators as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m mþ1 mþ1 m etðx þx @xÞ ¼ e x þðmþ1Þt xetx @x : ð3:4Þ Inserting this into (3.1), we obtain
X k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t mþ1 mþ1 m Tð mÞðxÞ¼e x þðmþ1Þt xetx @x Tð mÞðxÞ: k! nþk n kP0 Recalling from [15, Eq. (6.27)] that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m ekx @x f ðxÞ¼f mþ1 xmþ1 þðm þ 1Þk ; we have shown the following generalization of Proposition 2.1 to arbitrary order m.
Proposition 3.1. The Touchard polynomials of order m with m 2 N satisfy the relation
X k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t mþ1 mþ1 Tð mÞðxÞ¼e x þðmþ1Þt xTð mÞð mþ1 xmþ1 þðm þ 1ÞtÞ: ð3:5Þ k! nþk n kP0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mþ1 xmþ1þðmþ1Þt x ð mÞ According to [15, Theorem 6.14], e is the exponential generating function of the Tn ðxÞ. As in the case m ¼ 1 one can use this to derive from the disentanglement identity (3.4) the following generalization of Corollary 2.2 to arbi- trary order m.
Corollary 3.2. One has for m 2 N and k 2 N0 the operational relation Xk m m k k ð mÞ m r ðx þ x @xÞ ¼ Tk r ðxÞðx @xÞ : ð3:6Þ r¼0 r Similar to the case m ¼ 1, we can interpret this identity in terms of noncommuting variables U; V. In the case at hand we m m identify X # V as well as X @x # U and obtain from (3.3) the commutation relation
2mþ1 UV ¼ VU mV m ; ð3:7Þ T. Mansour, M. Schork / Applied Mathematics and Computation 219 (2013) 9978–9991 9983
2mþ1 which reduces to (2.12) for m ¼ 1. Thus, the case of order m corresponds to the choice of parameters h ¼ m and s ¼ m in the relation (2.11). The identity of Corollary 3.2 can be written in terms of the variables U; V as Xk k 1 k ð mÞ m r ðV þ UÞ ¼ Tk r ðV ÞU : ð3:8Þ r¼0 r In the case m ¼ 1 we could use an expression of the Touchard polynomials of order 1 in terms of Bessel polynomials to derive a nice binomial formula. In the case of arbitrary order m such an explicit expression is not known. However, using generalized Stirling numbers (or Bell polynomials) we can write (3.8) in another fashion. For this let us recall that general- n ized Stirling numbers Ss;hðn; kÞ were introduced in [13–15] as normal ordering coefficients of ðVUÞ in the variables U; V sat- isfying (2.11), i.e., one has
Xn n sðn kÞþk k ðVUÞ ¼ Ss;hðn; kÞV U : ð3:9Þ k¼0 The special case s ¼ 0 and h ¼ 1 corresponds to variables U; V satisfying UV ¼ VU þ 1 (i.e., the Weyl algebra) and it follows that S0;1ðn; kÞ¼Sðn; kÞ, the conventional Stirling numbers of the second kind. The corresponding generalized Bell polynomi- als are defined as usual, i.e.,
Xn k Bs;hjnðxÞ :¼ Ss;hðn; kÞx : k¼0 It was shown in [15, Theorem 6.8] that one has the relation
ð mÞ ðmþ1Þn T ðxÞ¼x Bmþ1 ðxÞ: ð3:10Þ n m ; mjn Therefore, we can write (3.8) as Xk k ðmþ1Þðk rÞ 1 k m m r ðV þ UÞ ¼ V Bmþ1 ðV ÞU : m ; mjk r r¼0 r This shows the following theorem.
2mþ1 Theorem 3.3. Let U; V be variables satisfying the commutation relation UV ¼ VU mV m with m 2 N. Then the following noncommutative binomial theorem holds true for any k 2 N0 Xk pffiffiffiffi k k k r ðmÞ m r ðV þ UÞ ¼ V Bk r V U ; ð3:11Þ r¼0 r
ðmÞ where the polynomial B‘ ðxÞ is given explicitly by X‘ ðmÞ ‘ 1 ‘ s B ðxÞ¼x Bmþ1 ðx Þ¼ Smþ1 ð‘; sÞx : ð3:12Þ ‘ m ; mj‘ m ; m s¼0 ðmÞ Theorem 3.3 is the generalization of Theorem 2.3 to arbitrary negative order m with m 2 N. The polynomials B‘ ðxÞ - which measure the influence of the noncommutativity of the variables U; V - seem to be rather poorly understood, in con- trast to the special case m ¼ 1 where Bessel polynomials appear.
k ðmþ1Þk 1 Remark 3.4. The highest power of V which appears in the normal ordering of ðU þ VÞ is V m (which reduces for m ¼ 1to V 2k 1). To see this, it is clear that in the sum of the right-hand side of (3.11) only the summand r ¼ 0 has to be considered, i.e., pffiffiffiffi pffiffiffiffi pffiffiffiffi k 1 k ðmÞ m ðmÞ m m V B V . From the explicit expression given in (3.12) and the fact that Smþ1 ð‘; kÞ¼d‘;k we find B V V , k m ; m k showing the assertion. This also fits the general formula given in (2.14). Note that on the right-hand side of (2.14) the highest kþðk 1Þðs 2Þ 2mþ1 power of V which appears is given in the case s > 2byV . In the case at hand we have s ¼ m > 2 and it follows kþðk 1Þðs 2Þ ðmþ1Þk 1 that V ¼ V m .
ðmÞ The same arguments as above can be used for the generalized Touchard polynomials Tn ðxÞ with m 2 N. Here we have [6, Eq. (39)] Xk m m k ðmÞ m r ðx þ x @xÞ¼ Tk rðxÞðx @xÞ : r¼0 r
2m 1 m # m # m As above, x V and x @x U corresponds to variables U; V satisfying UV ¼ VU þ mV . For these variables one has thus P k k k ðmÞ 1 r ðV þ UÞ ¼ T ðV mÞU . Using (4.2), this equals r¼0 r k r 9984 T. Mansour, M. Schork / Applied Mathematics and Computation 219 (2013) 9978–9991
Xk pffiffiffiffi ðk rÞ pffiffiffiffi k k k r m m r ðV þ UÞ ¼ V V Bm 1 V U : m ;mjk r r¼0 r Now, we can formulate the analog to Theorem 3.3.
2m 1 Theorem 3.5. Let U; V be variables satisfying the commutation relation UV ¼ VU þ mV m with m 2 N. Then the following noncommutative binomial theorem holds true for any k 2 N0 Xk pffiffiffiffi k k k r ðmÞ m r ðV þ UÞ ¼ V Ck r V U ; ð3:13Þ r¼0 r
ðmÞ where C‘ ðxÞ is given explicitly by X‘ ðmÞ ‘ s ‘ C ðxÞ¼x Bm 1 ðxÞ¼ Sm 1 ð‘; sÞx : ð3:14Þ ‘ m ;mj‘ m ;m s¼0
ðmÞ ðmÞ Remark 3.6. The function C‘ ðxÞ of Theorem 3.5 results from the polynomial B‘ ðxÞ appearing in Theorem 3.3 by switching formally from m to m as well as from x to x 1, i.e.,
ðmÞ ð mÞ 1 C‘ ðxÞ¼B‘ ðx Þ:
4. A recursion relation of the generalized Touchard polynomials
For the conventional Touchard polynomials one can easily derive a recursion relation due to the identification (1.2) with the Bell polynomials. Using that the exponential generating function of the Bell polynomials is given by exðez 1Þ, we obtain
X n z z T ðxÞ ¼ exðe 1Þ: n n! nP0 Taking a derivative with respect to z, this shows ! !() X n X l X m X Xn n z z z z n z T ðxÞ ¼ xezexðe 1Þ ¼ x T ðxÞ ¼ x T ðxÞ : nþ1 n! l! m m! k n! nP0 lP0 mP0 nP0 k¼0 k Comparing coefficients gives the recursion relation Xn n Tnþ1ðxÞ¼x TkðxÞ; ð4:1Þ k¼0 k which can be found in exactly this form in [2, Eq. (2.14)]. Now, we want to derive an analogous recursion relation for the ðmÞ generalized Touchard polynomials. For this we consider first the case Tn ðxÞ with m 2 N. Here we have the following relation to the generalized Bell polynomials [15, Theorem 6.1]
ðmÞ ðm 1Þn T ðxÞ¼x Bm 1 ðxÞ; ð4:2Þ n m ;mjn and the main task is to establish a recursion relation for the generalized Bell polynomials. In [14, Corollary 4.1] the exponen- tial generating function of the generalized Bell polynomials Bs;hjnðxÞ is given in a slightly implicit form. There one has to con- m 1 sider the cases s ¼ 0; s ¼ 1 and s 2 R nf0; 1g separately. In the case we are interested in we have sðmÞ¼ m , so that 1 sð1Þ¼0; sð2Þ¼2 and sðmÞ2ð0; 1Þ for m 2f2; 3; ...g. Since the case sðmÞ¼0 (i.e., m ¼ 1) is the conventional case, we restrict to the case m P 2. Then we can use [14, Corollary 4.1], giving the exponential generating function of the generalized Bell polynomials ÈÉ X n s 1 z x 1 ð1 hszÞ s B ðxÞ ¼ ehðs 1Þ : ð4:3Þ s;hjn n! nP0 Following the strategy from above, we take a derivative with respect to z and obtain ÈÉ X n s 1 X m z 1 x 1 ð1 hszÞ s 1 z B ðxÞ ¼ xð1 hszÞ s ehðs 1Þ ¼ xð1 hszÞ s B ðxÞ : s;hjnþ1 n! s;hjm m! nP0 mP0
Using ! X 1 X 1 r 1 r þ 1 r Cðr þ Þ r z ð1 hszÞ s ¼ s ðhsÞ zr ¼ s ðhsÞ ; 1 r! rP0 r rP0 CðsÞ T. Mansour, M. Schork / Applied Mathematics and Computation 219 (2013) 9978–9991 9985 one finds ÀÁ ! ! X zn X C r þ 1 zr X zm B ðxÞ ¼ x ÀÁs ðhsÞr B ðxÞ : s;hjnþ1 n! 1 r! s;hjm m! nP0 rP0 C s mP0 Comparing coefficients shows the following theorem.
Theorem 4.1. Let h – 0 and s 2 R nf0; 1g. The generalized Bell polynomials satisfy the recursion relation ÀÁ Xn n C n k 1 ÀÁþ s n k Bs;hjnþ1ðxÞ¼x 1 ðhsÞ Bs;hjkðxÞ: k¼0 k C s To obtain the recursion relation for the generalized Touchard polynomials, it remains to combine (4.2) and Theorem 4.1: Xn n Cðn k þ m Þ ðmÞ ðm 1Þðnþ1Þ m 1 ðm 1Þn ÀÁm 1 n k T ðxÞ¼x Bm 1;m n 1ðxÞ¼x x x ðm 1Þ Bm 1;m kðxÞ nþ1 m j þ k C m m j k¼0 m 1 Xn n Cðn k þ m Þ ¼ xm m 1 ½ðm 1Þxðm 1Þ n kTðmÞðxÞ: Cð m Þ k k¼0 k m 1 The last equation is the sought-for recursion relation in the case m P 2. Let us turn to the generalized Touchard polynomials ð mÞ of negative order, i.e., to Tn ðxÞ with m 2 N. Using (3.10), we see that the parameter sð mÞ of the corresponding generalized mþ1 Bell polynomial is given by sð mÞ¼ m 2ð1; 2 so that one can use Theorem 4.1 as in the case of positive order. A calculation similar to the one above shows that Xn m n C n k þ m 1 ð mÞ m þ ðmþ1Þ n k ð mÞ Tnþ1 ðxÞ¼x ½ ðm þ 1Þx Tk ðxÞ: k m k¼0 C mþ1
ðmÞ Note that this recursion relation is the same as the one which results from the one given for Tn ðxÞ above by switching for- mally from m to m! Thus, we can combine these two relations into one relation for arbitrary integer order.
Theorem 4.2. Let r 2 Z nf0; 1g. The generalized Touchard polynomials of order r satisfy the recursion relation ÀÁ Xn r r n C n k þ r Tð Þ ðxÞ¼xr ÀÁr 1 ½ðr 1Þxðr 1Þ n kTð ÞðxÞ: ð4:4Þ nþ1 C r k k¼0 k r 1
ð0Þ Remark 4.3. Let us discuss briefly the two cases r ¼ 0; 1 excluded in Theorem 4.2. In the trivial case r ¼ 0 one has Tn ðxÞ¼1 ð0Þ ð0Þ for all n, implying Tnþ1ðxÞ¼Tn ðxÞ. The case r ¼ 1 corresponds to the conventional case with recursion relation (4.1). Although we have excluded this case in the derivation of Theorem 4.2, we can nevertheless see what happens in (4.4) when 1 ‘‘r ! 1’’. Since there are terms containing r 1, we cannot insert r ¼ 1 directly. However, if we define for r 2 R with r > 1 ÀÁ C n k þ r ÀÁr 1 ðr 1Þ n k An;kðr; xÞ :¼ r ½ðr 1Þx ; C r 1 then we can write (4.4) as Xn ðrÞ r n ðrÞ Tnþ1ðxÞ¼x An;kðr; xÞTk ðxÞ: k¼0 k Since one has for fixed n; k; x that
limAn;kðr; xÞ¼1; r!1 this shows that – in a certain sense – (4.4) reduces to (4.1) for r ! 1. Before closing this section, let us point out that one can switch in the definition (1.1) to another variable as follows. If we 1 z d , dxðzÞ d d d let xðzÞ¼e , then one finds due to the transformation x dx xðzÞ dz dz that the operator x dx becomes the operator dz. Thus, one can write (1.1) equivalently as the well-known Rodriguez-like formula n z d z T ðezÞ¼e e ee : ð4:5Þ n dz
ðrÞ For Tn ðxÞ with arbitrary r 2 Z one can do the same thing (a closely related argument was used in [6]). Here we want to find a r d d , r d function xrðzÞ such that x dx transforms into dz. In general, one obtains by the change of variable x xrðzÞ for x dx the operator 9986 T. Mansour, M. Schork / Applied Mathematics and Computation 219 (2013) 9978–9991 1 r dxr ðzÞ d dxr ðzÞ r z fxrðzÞg dz dz so that we have to solve the differential equation dz ¼fxrðzÞg . For r ¼ 1 we obtain the solution x1ðzÞ¼e 1 r 1 ffiffiffiffiffiffiffiffiffiffi1 from above and are done. For r 2 Z nf1g we obtain the solution xrðzÞ¼fð1 rÞzg ¼ p and obtain as analog to (4.5) r 1 ð1 rÞz the following proposition.
ðrÞ Proposition 4.4. The generalized Touchard polynomials Tn satisfy for r 2 Z nf1g the Rodriguez-like formula ! n ffiffiffiffiffiffiffiffi1 ffiffiffiffiffiffiffiffi1 r 1 r p1 d r p1 Tð Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ e ð1 rÞz e ð1 rÞz: ð4:6Þ n r 1 ð1 rÞz dz
1 Example 4.5. Let us consider r ¼ 2. Here we get x2ðzÞ¼ z and, therefore, the nice formula n ð2Þ 1 1 d 1 T ¼ ez e z : n z dz pffiffiffiffiffi As another example we choose r ¼ 1. Here we get x 1ðzÞ¼ 2z and, consequently, ffiffiffiffi n ffiffiffiffi pffiffiffiffiffi p d p Tð 1Þ 2z ¼ e 2z e 2z: n dz
5. The Touchard functions of arbitrary real order
Recall that we defined for any m 2 Z nf0g the generalized Touchard polynomials of order m by
ðmÞ x m n x Tn ðxÞ¼e ðx @xÞ e ; (which for m < 0 are polynomials in x 1). It is then tempting to introduce for any a 2 R nf0g generalized Touchard functions of order a by exactly the same formula, i.e.,
ðaÞ x a n x Tn ðxÞ :¼ e ðx @xÞ e : ð5:1Þ a As above there exist two natural pairs of associated noncommutative variables. On the one hand one can consider V 1 ¼ X a 1 a a 1 and U1 ¼ @x, implying the commutation relation U1V 1 ¼ V 1U1 þ aV 1 . Thus, this gives an example of (2.11) with s ¼ a and h ¼ a. It follows from (3.9) that Xn a 1 n a ðn kÞþk k ðV 1U1Þ ¼ Sa 1 ðn; kÞV U ; a ;a 1 1 k¼0 implying Xn ðaÞ ða 1Þn k ða 1Þn T ðxÞ¼x Sa 1 ðn; kÞx ¼ x Ba 1 ðxÞ: ð5:2Þ n a ;a a ;ajn k¼0 2a 1 a a a On the other hand, one can consider the pair U2 ¼ X @x and V 2 ¼ X with the commutation relation U2V 2 ¼ V 2U2 þ aV 2 . a a k Thus, if we want to consider ðx þ x @xÞ in analogyÀÁ to above, we are led to the pair fV 2; U2g of noncommuting variables 2a 1 which is a particular case of (2.11) with ðsa; haÞ¼ a ; a . ÀÁ 1 1 Example 5.1. For the choice a ¼ one finds ðs1; h1Þ¼ 0; , i.e., the variables V2 and U2 satisfy the (scaled) Weyl algebra 2 2 2 2 1 k U2V 2 ¼ V2U2 þ 2. In this case the explicit formula for ðV2 þ U2Þ has been established already long ago, see the remarks and literature given in [13]. To be more concrete, consider the operators X and Dh (with h 2 R) satisfying the commutation relation DhX¼XDh þ h. Then one can write Xk k k r ðX þ DhÞ ¼ Hk rðX; hÞDh; ð5:3Þ r¼0 r where the polynomials Hmðx; hÞ are a variant of the Hermite polynomials, see the discussion in [13]. In an equivalent form this identity was already known to Burchnall [3].In[9] a variant of the BurchnallR identity was discussed in the context of ^ x generalized shift operators E^ ¼ ekqðxÞðd=dxÞ ¼ ekT x with T^ ¼ qðxÞ d . Defining F ðxÞ¼ df , one obtains the identity x dx q qðfÞ Xk ^ k k r ð2Þ ^ r ðFqðxÞþ2yT xÞ ¼ ð2yÞ hk rðx; yÞðT xÞ ; r¼0 r ð2Þ where the functions hm ðx; yÞ are called pseudo-Hermite–Kampé de Feriet polynomials [9]. Here one has the commutation ^ ^ relation ½2yT x; FqðxÞ ¼ y, i.e., FqðxÞ and 2yT x satisfy a (scaled) Weyl algebra. Choosing y ¼ 1 and qðxÞ¼1, i.e., FqðxÞ¼x ^ d and T x ¼ dx, one obtains the conventional Burchnall identity. T. Mansour, M. Schork / Applied Mathematics and Computation 219 (2013) 9978–9991 9987
Comparing the identity mentioned in the preceding example with (3.8), one expects that the Touchard functions of order 1 a ¼ 2 are given by Hermite polynomials. This is what we will show in Theorem 5.3 after having established a necessary Lemma.
Lemma 5.2. Let h 2 C nf0g. The generalized Bell polynomials satisfy x B ðxÞ¼hnB : s;hjn s;1jn h
n k Proof. Using the relation Ss;hðn; kÞ¼h Ss;1ðn; kÞ of the generalized Stirling numbers [14], the assertion follows directly from the definition of the generalized Bell polynomials since Xn Xn x k x B ðxÞ¼ S ðn; kÞxk ¼ hn S ðn; kÞ ¼ hnB ; s;hjn s;h s;1 h s;1jn h k¼0 k¼0 as requested. h
Recall that the Hermite polynomials HnðxÞ can be defined by their exponential generating function [5, p. 50], X n 2 t e2tz t ¼ H ðzÞ : ð5:4Þ n n! nP0
Theorem 5.3. The Touchard functions of order 1 can be expressed by Hermite polynomials, i.e., 2 n ðÞ1 i ÀÁpffiffiffi T 2 ðxÞ¼ H i x : ð5:5Þ n 2 n
Proof. From (5.2) we immediately obtain 1 n ðÞ2 n 1 2 ffiffiffi Tn ðxÞ¼x B 1;1jnðxÞ¼ p B 1;1jnð2xÞ; 2 2 x where we have used Lemma 5.2 in the second equation. In [15, Proposition 6.13] it was shown that pffiffiffi n pffiffiffi i ffiffiffiy yffiffiffi B 1;1jnðyÞ¼ p Hn p : 2 i 2 Inserting this into the above equation yields the assertion. h
1 ðÞ2 Remark 5.4. The connection between Tn ðxÞ and HnðxÞ can also be seenÀÁ as follows. Using that the formula given in Propo- 1 z 2 sition 4.4 also holds for r 2 R nf1g, we can choose r ¼ 2 to find x1=2ðzÞ¼ 2 and, therefore, 2 n 1 z 2 z 2 ðÞ2 z d T ¼ e ðÞ2 eðÞ2 : n 2 dz ÀÁ n x2 d n x2 Recalling that the classical Rodriguez formula for HnðxÞ is given by HnðxÞ¼ð 1Þ e dx e [17, p. 45], this shows the con- nection in an alternative way. 2a 1 Above we have seen that the variables U2; V 2 corresponding to a satisfy (2.11) with sa ¼ a (and ha ¼ a). Let us check 1 1 1 1 when sa 2 Z nf2g.Ifsan ¼ n then an ¼ 2 n. Thus, a0 ¼ 2 ; a1 ¼ 1; a3 ¼ 1; a4 ¼ 2 ; a5 ¼ 3 ; ..., whereas 1 1 1 1 1 d a 1 ¼ 3 ; a 2 ¼ 4 ; a 3 ¼ 5 ; ... For example, if a ¼ 1 then fx ; x dxg corresponds to variables fV 2; U2g satisfying 3 1=3 1=3 d U2V 2 ¼ V 2U2 V 2 as discussed in Section 2.Ifa ¼ 1=3 then fx ; x dxg corresponds to variables fV 2; U2g satisfying 1 5 U2V 2 ¼ V 2U2 3 V 2. To derive a binomial formula for these variables in analogy to above one needs information about, ðaÞ e.g., the exponential generating function of Tn ðxÞ. However, since we cannot cite the relevant properties (as in the case of order m with m 2 Z) we refrain from a closer study of these functions and turn instead to another generalization in the next section.
Remark 5.5. Let us point out that in [1] certain analogs of the Touchard (or exponential) polynomials are considered. ÀÁ P P n k Recalling the definition T ðxÞ¼e x x d ex ¼ n Sðn; kÞxk, one can consider instead of ex ¼ x the function P n dx k¼0 kP0 k! 1 k 1 x ¼ kP0x and define in analogy to TnðxÞ the functions d n 1 U ðxÞ :¼ð1 xÞ x : n dx 1 x 9988 T. Mansour, M. Schork / Applied Mathematics and Computation 219 (2013) 9978–9991 P ÀÁÈÉ ÀÁ n k d n 1 1 x Introducing the geometric polynomials by xnðxÞ¼ k¼0Sðn; kÞk!x [1, Eq. (3.3)], one can show that x dx 1 x ¼ 1 x xn 1 x [1, Eq. (3.8)]. Thus, x U ðxÞ¼x : n n 1 x ÀÁ n x If we denote by AnðxÞ the Eulerian polynomials [5], one has AnðxÞ¼ð1 xÞ xn 1 x [1, Eq. (3.18)], implying n UnðxÞ¼ð1 xÞ ÀÁAnðxÞ.ÈÉ Following the idea of the present section, one should define for a 2 R the functions ðaÞ a d n 1 Un ðxÞ :¼ð1 xÞ x dx 1 x and consider their properties.
6. Outlook: a further generalization of Touchard functions
In this section we want to sketch a possible further generalization of theÀÁ Touchard functions considered above. Recall that ðaÞ ðaÞ x a d n x a d we defined Touchard functions Tn ðxÞ for arbitrary a 2 R by Tn ðxÞ¼e x dx e . The operator x dx can be interpreted in an algebraic fashion as derivation (and in a more geometric fashion as a vector field). Its exponential thus represents an auto- morphism (and is also called generalized shift operator [9] or exponential operator [8]). A general derivation can be written in the form d T^ gðxÞ ; x dx where the function g is assumed to be ‘‘sufficiently smooth’’ (e.g., analytic in an open interval). Using such a general deriva- a d tion instead of x dx, one is led to the following definition.
ðgÞ Definition 6.1 (Comtet–Touchard function associated to g). Let g be a smooth function. The Comtet–Touchard functions Tn ðxÞ associated to g are defined for n 2 N by d n TðgÞðxÞ :¼ e x gðxÞ ex: ð6:1Þ n dx Clearly, for gðxÞ¼xa this definition gives back the Touchard functions considered in previous sections. The reason for call- d n ing these functions Comtet–Touchard functions is that L. Comtet considered in 1973 [4] expressions of the form ðgðxÞ dx Þ in detail and obtained in particular the following theorem.
Theorem 6.2 (Comtet). Let g be a smooth function. Then one has for any n 2 N the expansion d n Xn d l gðxÞ ¼ TðgÞðxÞ ; dx n;l dx l¼1 where for 1 6 l 6 n X Yn 1 g ðxÞ ðgÞ gðxÞ kj Tn;l ðxÞ¼ ðj þ 1 k1 kjÞ ; l! kj! k1þ þkn 1¼n lðkiP0Þ j¼1 6 6 k1þ þki ið1 i\nÞ where g ðxÞ is the k -th derivative function of gðxÞ. kj j ðgÞ The numbers appearing as coefficients in the above expressions for Tn;l ðxÞ are related to the number of rooted trees and can be found as A139605 in OEIS [18]. Let us give the cases n ¼ 2; 3 explicitly where we write briefly g ¼ gðxÞ: d 2 d d 2 g ¼ gg þ g2 ; dx 1 dx dx d 3 d d 2 d 3 g ¼ðgg2 þ g2g Þ þ 3g2g þ g3 : dx 1 2 dx 1 dx dx
ðgÞ ðgÞ ðgÞ 2 Thus, one has for n ¼ 1 trivially T1;1 ¼ g, for n ¼ 2 that T2;l ¼ gg1; T2;2 ¼ g and for n ¼ 3 that ðgÞ 2 2 ðgÞ 2 ðgÞ 3 ðgÞ n T3;1 ¼ gg1 þ g g2; T3;2 ¼ 3g g1; T3;3 ¼ g . It is easy to see that one has in general Tn;n ¼ g .
Proposition 6.3. The Comtet–Touchard functions associated to g are given for any n 2 N by
Xn ðgÞ ðgÞ Tn ðxÞ¼ Tn;l ðxÞ; ð6:2Þ l¼1
ðgÞ where Tn;l ðxÞ are given by Theorem 6.2.
Proof. Inserting the expansion given in Theorem 6.2 into (6.1) shows the assertion. h T. Mansour, M. Schork / Applied Mathematics and Computation 219 (2013) 9978–9991 9989
Let us give the first few Comtet–Touchard functions explicitly where we denote the derivative with respect to x by a prime:
ðgÞ T1 ðxÞ¼gðxÞ; ðgÞ 0 T2 ðxÞ¼gðxÞfg ðxÞþgðxÞg; ðgÞ 0 2 00 0 2 T3 ðxÞ¼gðxÞfðg ðxÞÞ þ gðxÞg ðxÞþ3gðxÞg ðxÞþgðxÞ g: If we follow the procedure of the previous sections, we should consider the exponential generating function of the Comtet– Touchard functions. Using the operational method, one obtains from the definition ! X n X n n X n n t ðgÞ t x d x x t d x x tgðxÞ d x T ðxÞ¼ e gðxÞ e ¼ e gðxÞ e ¼ e e dxe : n! n n! dx n! dx nP0 nP0 nP0 Following [8,9] (see also [11]), we introduce Z x df F ðxÞ :¼ ; g gðfÞ
1 and denote by Fg its inverse. Then one can write [8, Eq. (I.2.7)] for the associated exponential operator
tg x d 1 ð Þdx e f ðxÞ¼f fFg ðFgðxÞþtÞg: ð6:3Þ Thus, we have shown the following proposition.
Proposition 6.4. The exponential generating function of the Comtet–Touchard functions associated to g is given by
X n t ðgÞ 1 T ðxÞ¼eFg fFg ðxÞþtg x; ð6:4Þ n! n nP0 R x df 1 where FgðxÞ :¼ gðfÞ and Fg denotes its inverse. The expression given in (6.4) hides the complexity that one has to solve an integral for Fg and find the inverse function 1 a Fg . Clearly, if we choose gðxÞ¼x (in particular with a 2 Z) we obtain the results discussed in previous sections. In the gen- eral framework one could also be interested in other cases, for example polynomials instead of monomials. R 2 x df 1 Example 6.5. Let us consider gðxÞ¼x þ 1. It follows that FgðxÞ¼ 2 ¼ arctanðxÞ as well as F ðxÞ¼tanðxÞ. Thus, from 1þf g (6.4) one finds X tn TðgÞðxÞ¼etanfarctanðxÞþtg x: n! n nP0
Using tanða þ bÞ¼ tanðaÞþtanðbÞ , one can get rid of the arctan and obtains 1 tanðaÞ tan ðbÞ X n t ð1 þ xÞ tanðtÞ TðgÞðxÞ¼exp : n! n 1 x tan t nP0 ð Þ In the same fashion one can use (6.3) to obtain for the action of the associated exponential operator tð1þx2Þ d x cosðtÞþsinðtÞ e dxf ðxÞ¼f ; cosðtÞ x sinðtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a relation which was already given in [8, Eq. (I.2.7)]. Analogous formulas for the case gðxÞ¼ x2 þ 1 can also be found in [8]. It is worth pointing out that in [8] a wealth of information concerning exponential operators and operational rules can be found, e.g., when the exponent is generalized to an arbitrary first order differential operator or to (special) operators contain- ing higher order derivatives. Ihara considered similar problems in a closely related context in a more algebraic fashion [11].A generalization of Theorem 6.2 to the case of several dimensions has been established by Ginocchio [10]. In the following re- ðgÞ mark the consideration of Tn ðxÞ from above is generalized further, but only in principle (due to the increasing complexity it will be extremely difficult to obtain explicit expressions).
Remark 6.6. Recall that we defined the Comtet–Touchard functions associated to g by (6.1). Here we have used the d derivation gðxÞ dx associated to g. More generally, we can also consider a general differential operator of order one, i.e., d Dg;v ¼ gðxÞ dx þ vðxÞ. Thus, we introduce the following Comtet–Touchard functions associated to ðg; vÞ by n ðg;vÞ x n x x d x T ðxÞ :¼ e ðDg; Þ e ¼ e gðxÞ þ vðxÞ e : ð6:5Þ n v dx 9990 T. Mansour, M. Schork / Applied Mathematics and Computation 219 (2013) 9978–9991
dm d It is clear that one can generalize this to differential operators Da ¼ amðxÞ dxm þ þa1ðxÞ dx þ a0ðxÞ (with a x n x a ¼ðam; am 1; ...; a1; a0Þ) of arbitrary order by setting TnðxÞ :¼ e ðDaÞ e . However, since the resulting expressions will be- come quickly messy, we restrict to the first order case. We will not attempt to give an explicit expression for d n fgðxÞ dx þ vðxÞg , thereby generalizing Theorem 6.2 of Comtet. Instead, we consider the exponential generating function of ðg;vÞ Tn ðxÞ and the same argument as above shows that X n t ðg;vÞ x tfgðxÞ d þvðxÞg x T ðxÞ¼e e dx e : n! n nP0
The action of the exponential operator can be described as follows [8, p. 6]. If we let xðtÞ and kðtÞ be the solutions of the sys- tem of first order differential equations d xðtÞ¼gðxðtÞÞ; xð0Þ¼x; dt d kðtÞ¼vðxðtÞÞkðtÞ; kð0Þ¼1; dt then one has the relation [8, Eq. (I.2.25)]:
tfgðxÞ d þvðxÞg e dx f ðxÞ¼f ðxðtÞÞkðtÞ: ð6:6Þ (The notation is slightly misleading since kðtÞ can also depend on x, see the example below.) Note that in the case v ¼ 0 the 1 second differential equation implies kðtÞ¼1 for all t, so that we get back the result (6.3), i.e., xðtÞ¼Fg fFgðxÞþtg in the nota- tion from above. Thus, using these notations, we can write X n t ðg;vÞ xðtÞ x F 1fF ðxÞþtg x T ðxÞ¼e kðtÞ¼e g g kðtÞ: n! n nP0
d As a particular example, let us consider the operator dx þ x, i.e., gðxÞ¼1 as well as vðxÞ¼x. The first differential equation reduces to x0ðtÞ¼1 with xð0Þ¼x, yielding xðtÞ¼x þ t. Therefore, the second differential equation reduces to k0ðtÞ¼ðx þ tÞkðtÞ with kð0Þ¼1, yielding kðtÞ¼etxþt2=2. Thus,
X n 2 t ð1;idÞ t txþt T ðxÞ¼e e 2 : n! n nP0 The right-hand side is very similar to the exponential generating function of the Hermite polynomials, see (5.4). This is not surprising since we can use the Burchnall identity (5.3) to obtain ! d n Xn n d r Xn n Tð1;idÞðxÞ¼e x þ x ex ¼ e x H ðx; 1Þ ex ¼ H ðx; 1Þ: n dx n r dx n r r¼0 r r¼0 r
7. Conclusions
We discussed several properties of the generalized Touchard polynomials, e.g., a recursion relation generalizing the one of the conventional Touchard polynomials. Furthermore, an interpretation of some operational formulas was given in terms of a binomial theorem for particular noncommuting variables. It was suggested to consider generalized Touchard functions a d associated to x dx with arbitrary a 2 R and as a first example it was shown that the resulting Touchard functions of order a ¼ 1=2 are given by Hermite polynomials. Generalizing still further, we introduced so called Comtet–Touchard functions d associated to arbitrary derivations gðxÞ dx and showed first properties of these functions, using in particular Comtet’s result d about powers of gðxÞ dx (hence the name Comtet–Touchard). We believe that the class of Comtet–Touchard functions pro- vides a natural and unifying framework which merits closer study. Let us mention some possible avenues for future investigations. Apart from the already mentioned closer study of the Comtet–Touchard functions, we think it might be worthwhile to consider a q-deformed version of the generalized Touchard polynomials. Let us give some details. If we denote by Sqðn; kÞ the q-deformed Stirling numbers ofP the second kind and by Dq f ðqxÞ f ðxÞ n n k k the Jackson-derivative Dqf ðxÞ¼ qðx 1Þ , one has the well-known operational relation ðXDqÞ ¼ k¼0Sqðn; kÞX Dq. Using the 2 n 1 basic numbers ½n q ¼ 1 þ q þ q þ þq , we introduce the two classic q-deformed exponential functions X X n xn qðÞ2 xn e ðxÞ¼ ; E ðxÞ¼ : q n ! q n ! nP0 ½ q nP0 ½ q
1 Recalling that eq is an eigenfunction for Dq and that Eqð xÞ¼ðeqðxÞÞ , we may introduce the q-deformed generalized Tou- chard polynomials of order m 2 Z (and n 2 N)by
ðmÞ m n Tn;q ðxÞ :¼ Eqð xÞðx DqÞ eqðxÞ: T. Mansour, M. Schork / Applied Mathematics and Computation 219 (2013) 9978–9991 9991
Clearly, considering q ¼ 1 gives back the generalized Touchard polynomialsP studied in the present paper. For m ¼ 1 we write ð1Þ n k Tn;qðxÞ¼Tn;qðxÞ and obtain in close analogy to (1.2) that Tn;qðxÞ¼ k¼0Sqðn; kÞx ¼ Bn;qðxÞ. One may then hope that it is pos- sible to generalize the results of the present paper to this q-deformed situation. ðmÞ In a different direction, the functions B‘ ðxÞ appearing in Theorem 3.5 and describing the influence of the noncommuta- 2m 1 tivity of the variables U; V (satisfying UV ¼ VU mV m ) in the binomial formula should be understood better. In the case ðmÞ m ¼ 1 the same role is played by the Bessel polynomials, so the functions B‘ ðxÞ should be considered as some kind of ‘‘high- er order analog’’ to the Bessel polynomials. The coefficients of the Bessel polynomials – the Bessel numbers – have a nice ðmÞ combinatorial interpretation, so one should consider the coefficients of B‘ ðxÞ in an analogous fashion and find out whether they have a nice combinatorial interpretation, too. As a final point we would like to mention the consideration of the generalized Touchard functions of order a 2 R. We con- 1 ðÞ2 sidered the particular case a ¼ 1=2 and showed that Tn ðxÞ is given by a Hermite polynomial. It would be interesting to find out whether one has a relation to other well-known polynomials for appropriate choices of a. Furthermore, one should also consider the associated exponential generating function, the recursion relation, etc.
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