Multivariate generating functions built of Chebyshev polynomials and some of its applications and generalizations
aa r X i v : . [ m a t h . C A ] D ec MULTIVARIATE GENERATING FUNCTIONS INVOLVINGCHEBYSHEV POLYNOMIALS
PAWE L J. SZAB LOWSKI
Abstract.
We sum multivariate generating functions composed of mixturesof Chebyshev polynomials of the first and second kind. That is, we find closedform of expressions of the type P j ≥ ρ j Q km =1 T j + t m ( x m ) Q n + km = k +1 U j + t m ( x m ) , for different integers t m , m = 1 , ..., n + k. We also sum Kibble-Slepian formulaof n variables with Hermite polynomials replaced by Chebyshev polynomialsof the first and the second kind. In all those cases that were considered, theobtained closed forms are rational functions with positive denominators. Weperform all these calculations basically in order to simplify certain multivariateintegrals or to obtain multivariate distribution having compact support as wellas for pure curiosity. We also hope that the obtained formulae will be useful inthe so-called free probability. We expect also that the obtained results wouldinspire further research and generalizations. In particular that following meth-ods presented in this paper one would be able to obtain formulae for the socalled q − Hermite polynomials. All those hopes are based on the observationthat the Chebyshev polynomials of the second kind considered here are the q -Hermite polynomials for q = 0. Introduction
In this note is we obtain close forms of the following expressions:I. The multivariate generating functions:(1.1) χ ( t ,...,t k + n ) k,n ( x ...x n + k | ρ ) = X j ≥ ρ j k Y m =1 T j + t m ( x m ) n + k Y m = k +1 U j + t m ( x m ) , where | t m | , k, n, j ∈ { , , ... } , k + n ≥ , | ρ | < , | x m | ≤ , m = 1 , ..., k + n and T j , U j denote j − th Chebyshev polynomials respectively of the first and second kind.II. The so-called Kibble–Slepian formula for Chebyshev polynomials i.e. closeforms of the expressions: f T ( x | K n ) = X S ( Y ≤ i Mathematics Subject Classification. Primary 42C10, 33C47, Secondary 26B35 40B05. Key words and phrases. multivariate generating functions, Kibble-Slepian formula, Chebyshevpolynomials. entries of the matrix S n a symmetric n × n matrix with zeros on the main diagonaland entries s ij being nonnegative integers, while s m denotes the sum of entries s ij along the m − th row of the matrix S n . Notice that Chebyshev polynomials of the second kind have played a similar rolein recently rapidly developing ”free probability” as the Hermite polynomials playin classical probability. Hence the results presented below are of significance for thefree probability theory.There is one more reason for which the results are important. Namely, sinceChebyshev polynomials of the second kind are identical with the so-called q − Hermitepolynomials for q = 0 . Hence the results of the paper can be an inspiration to obtainsimilar results for the q − Hermite polynomials and also to obtain ’ q − generalization’of the so-called Chebyshev polynomials of the first kind. All these ideas are ex-plained and made more precise in the sequence of observations, remarks, hypothesisand conjectures presented in section 5, below. We present there also an interestingnontrivial example of application of the method presented in Theorem 1 applied towell known cases and leading to non-obvious identities.The paper is organized as follows. In the next section we present some elementaryobservations, recall basic properties of Chebyshev polynomials as well as we provesome important auxiliary results. The main results of the paper are presentedin the two successive sections 3 and 4 presenting respectively closed form of one-parameter multivariate characteristic functions and the close form of the analog ofKibble–Slepian formula. The next section 5 presents generalization, observationsconjectures and examples. Finally the last section 6 presents longer proofs.2. Auxiliary results and elementary observations We have the following immediate observation. Remark 1. Since ∀ n ∈ N : T n (1) = 1 we have for all k ≥ χ ( t ,...,t k + n ) k,n (1 , x ...x n + k | ρ ) = χ ( t ,...,t k + n ) k − ,n ( x ...x n + k | ρ ) . We will show that in case I. all functions χ k,n are rational with common denom-inator w n + k ( x, ..., x k + n | ρ ) which is a symmetric polynomial in x , ..., x n + k of order2 n + k − as well as in ρ of order 2 n + k .In case II. also both functions f T ( x | K n ) and f U ( x | K n ) are rational with thesame denominator V n ( x | K n ) = Q n − j =1 Q nk = i +1 w ( x i , x j | ρ ij ), where w is defined by(2.7).Both statements will be proved in the sequel. The first one in Section 3 and thesecond in Section 4.The possible applications of the results of the paper seem, for example, to help(1) to simplify calculations of multiple integrals of the form Z ... Z k fold v m ( x , ..., x n | p )Ω n ( x , ..., x n | p ) k Y j =1 (1 − x i ) m j / dx ...dx k , where v m denotes some polynomial in variables x , ..., x n and numbers m j ∈ {− , } , p denotes a set of parameters that is might be different in casesI. or II. Then Ω n is equal to w n in case I. or V n in case II.(2) to obtain some expansions of the form related to (1.3) and (1.2) by puttingfixed values of some variables x j , ULTIVARIATE GENERATING FUNCTIONS 3 (3) to obtain families of multivariate distributions in R n with compact supportof the form: f n ( x , ..., x n ) = p m ( x , ..., x n | p )Ω n ( x , ..., x n | p ) n Y j =1 (1 − x i ) m j / , where polynomial p m can depend on many parameters can have any degreebut must me positive on S = n O j =1 [ − , 1] and such that f n integrates to 1 on S . Recall also that we have:(2.1) U n (cos α ) = sin(( n + 1) α ) / sin( α ) and T n (cos α ) = cos( nα )and that Z − T i ( x ) T j ( x ) 1 π √ − x dx = if i = j / if i = j = 01 if i = j = 0 , Z − U i ( x ) U j ( x ) 2 π p − x dx = (cid:26) if i = j if i = j . Immediately we have the following properties of the functions : Proposition 1. For n ≥ χ ( t ,...,t k ,t k +1 ,..., t n + k − ,t n + k ) k,n ( x , ..., x k , ..., x k + n − , | ρ )= ∂∂ρ χ t − ,...,t k − ,t k + n − − k,n − ( x , ..., x k , ..., x k + n − | ρ )+ t n + k φ t ,...,t n − n − ( x , ..., x k , ..., x k + n − | ρ ) . Proof. The identity follows the fact that U n (1) = n + 1 . Consequently we have: χ t ,...,t k ,.,..t k + n − ,t k + n k,n ( x , ..., x k , ., x k + n − , | ρ )= X j ≥ ( j + 1 + t n + k ) ρ j k Y s =1 T j + t s ( x s ) k + n − Y s = k +1 U j + t s ( x s )= ∂∂ρ X j ≥ ρ j +1 k Y s =1 T j + t s ( x s ) k + n − Y s = k +1 U j + t s ( x s )+ t n + k X j ≥ ρ j k Y s =1 T j + t s ( x s ) k + n − Y s = k +1 U j + t s ( x s ) . (cid:3) In the sequel we will use the convention that if all integer parameters t , ..., t n + k are equal zero then we will drop them. Notice that the functions φ and ϕ are known PAWE L J. SZAB LOWSKI for n = 1 and n = 2 and t = 0 , t = 0. Namely we have: χ , ( x | ρ ) = 1 w ( x | ρ ) ; χ , ( x | ρ ) = 1 − ρxw ( x | ρ ) , (2.2) χ , ( x, y | ρ ) = X n ≥ ρ n U j ( x ) U j ( y ) = 1 − ρ w ( x, y | ρ ) , (2.3) χ , ( x, y | ρ ) = X n ≥ ρ n T j ( x ) U j ( y ) = (1 − ρ + 2 ρ (cid:0) x + y (cid:1) − (cid:0) ρ + 3 (cid:1) ρxy ) w ( x, y | ρ ) , (2.4) χ , ( x, y | ρ ) = X n ≥ ρ n U n ( x ) T n ( y ) = (1 − ρ − ρxy + 2 ρ y ) w ( x, y | ρ ) , (2.5)where above and in the sequel we use the following bivariate, symmetric polynomialof order 1 and 2 : w ( x | ρ ) = 1 − ρx + ρ , (2.6) w ( x, y | ρ ) = (1 − ρ ) − xyρ (1 + ρ ) + 4 ρ ( x + y ) . (2.7)Notice also that both χ , and χ , are positive on [ − , × [ − , . Recall thatthe formulae in (2.2) are well known. The first formula in (2.3) it is famous Poisson-Mehler formula for q − Hermite polynomials where we set q = 0 . The second formulain (2.4) and in (2.5) have been recently obtained in [8].To calculate functions χ ( t ,...,t k + n ) k,n swiftly we need the following auxiliary results.Let us notice immediately that we have: Proposition 2. (2.8) w (cos( α + β ) | ρ ) w (cos( α − β ) | ρ ) = w (cos( α ) , cos( β ) | ρ ) . Proof. We have(1 − ρ cos( α + β ) + ρ )((1 − ρ cos( α − β ) + ρ ) =(1 + ρ ) − ρ (1 + ρ )(cos( α + β ) + cos( α − β )) + 4 ρ cos( α + β ) cos( α − β ) . Now recall that cos( α + β ) + cos( α − β ) = 2 cos( α ) cos( β ) and cos( α + β ) cos( α − β )= cos α + cos β − . (cid:3) Proposition 3. n Y j =1 cos( α j ) = 12 n X i ∈{− , } ... X i n ∈{− , } cos( n X k =1 i k α k ) , (2.9) n Y j =1 sin( α j ) n + k Y j = n +1 cos( α j ) = ( − ( n +1) / n + k P i ∈{− , } ... P i n + k ∈{− , } ( − P nl =1 ( i l +1) / sin( P n + kl =1 i l α l ) if n is odd ( − n/ n + k P i ∈{− , } ... P i n + k ∈{− , } ( − P nl =1 ( i l +1) / cos( P n + kl =1 i l α l ) if n is even . (2.10) Proof. Is shifted to section 6 (cid:3) ULTIVARIATE GENERATING FUNCTIONS 5 Lemma 1. Let n ∈ N , | ρ i | < , α i ∈ R , i ∈ S n = { , ..., n } . Let M i,n denote asubset of the set S n containing i elements. Let P M i,n ⊆ S n denote summation overall M i,n contained in S n . We have: X k ≥ ... X k n ≥ ( n Y i =1 ρ k i i ) cos( β + n X i =1 k i α i ) =(2.11) P nj =0 ( − j P M j,n ⊆ S n ( Q k ∈ M j,n ρ k ) cos( β − P k ∈ M j,n α k ) Q ni =1 (1 + ρ i − ρ i cos( α i )) , X k ≥ ... X k n ≥ ( n Y i =1 ρ k i i ) sin( β + n X i =1 k i α i ) =(2.12) P nj =0 ( − j P M j,n ⊆ S n ( Q k ∈ M j,n ρ k ) sin( β − P k ∈ M j,n α k ) Q ni =1 (1 + ρ i − ρ i cos( α i )) . We will need also the following special cases of formulae (2.11) and (2.12). Wewill formulate them as corollary. Corollary 1. For all | ρ | < we have X n ≥ ρ n sin( nα + β ) = (sin( β ) − ρ sin( β − α )) / (1 − ρ cos( α ) + ρ ) , (2.13) X n ≥ ρ n cos( nα + β ) = (cos( β ) − ρ cos( β − α ) / (1 − ρ cos( α ) + ρ ) . (2.14) Proof. We consider (2.12) and (2.11) and set n = 1 and α = α . (cid:3) One parameter sums. Multivariate generating functions ofChebyshev polynomials Now we have the following theorem: Theorem 1. For all integers n, k ≥ , | x s | < , t s ∈ Z , s = 1 , ..., n + k, we have: χ ( t ,...,t n + k ) k,n ( x , ..., x n + k | ρ ) = X j ≥ ρ j k Y s =1 T j + t s ( x s ) n + k Y s =1+ k U j + t s ( x s )(3.1) = l ( t ,...,t n + k ) k,n ( x , ..., x n + k | ρ ) w n + k ( x , ..., x n + k | ρ ) , (3.2) where w m ( x , ..., x m | q ) is a symmetric polynomial of order m − in x ..., x m andof order m in ρ defined by the following recurrence with w ( x | q ) given by (2.6) : w m +1 ( x , ..., x m − , x m , x m +1 | ρ ) =(3.3) w m ( x , ..., x m − , cos( α + β ) | ρ ) w m ( x , ..., x m − , cos( α + β ) | ρ ) | x m =cos( α ) x m +1 =cos( β ) ,n ≥ . PAWE L J. SZAB LOWSKI l n,k ( x , ..., x n + k | q ) is another polynomial given by the relationship: l ( t ,...,t n + k ) k,n ( x , ..., x n + k | ρ ) =(3.4) n + k − X j =0 ρ j j X m =0 m ! d m dρ m w k + n ( x , ..., x k + n | ρ ) (cid:12)(cid:12)(cid:12)(cid:12) ρ =0 × k Y s =1 T ( j − m )+ t s ( x s ) n + k Y s =1+ k U ( j − m )+ t s ( x s ) . (3.5) Corollary 2. For n ≥ we have: w n (1 , ...x n − , x n | ρ ) = ( w n − ( x , ...x n | ρ )) In particular w ( x , cos( α ) , cos( α ) | ρ ) = w ( x , cos( α + α ) | ρ ) w ( x , cos( α − α ) | ρ ) , which after replacing cos( α ) − > x and cos( α ) − > x and with the help ofMathematica yields: w ( x , x , x | ρ ) = 16 ρ ( x + x + x ) − ρ (1 + ρ ) ( x + x + x )(3.6)+16 ρ (1 + ρ )( x y + x z + y z ) + 64 ρ x y z − ρ (1 + ρ ) xyz ( x + x + x ) − ρ (1 + ρ )(1 + ρ − ρ ) xyz + (1 + ρ ) . Remark 2. Notice that from Theorem 1 we deduce that for all integers t , ..., t k + n the ratio χ ( t ,...,t k + n ) k,n ( x ...x n + k | ρ ) χ (0 ,..., k,n ( x ...x n + k | ρ ) is a rational function of arguments x , ..., x n + k , ρ. Such observation for was first made by Carlitz for k + n = 2 , and nonnegativeintegers t and t concerning the so called Rogers–Szeg¨o polynomials and two vari-ables x and x in [1] (formula 1.4). Later it was generalized by Szab lowski for theso called q − Hermite polynomials also for two variables in [3] . Now it turns out thatfor q = 0 the q − Hermite polynomials are equal to Chebyshev polynomials of thesecond kind hence one can state that so far the above mentioned observation wasknown for k = 0 and n = 2 . Hence we deal with far reaching generalization both inthe number of variables as well as for the Chebyshev polynomials of the first kind. Corollary 3. For | x i | ≤ and | ρ | < , n ≥ χ n, ( x , ..., x n | ρ ) ≥ , Z − ... Z − ( j times n Y s =1 π p − x s ) χ n, ( x , ..., x n | ρ ) dx ...dx j = n Y s = j +1 π p − x s , for j = 1 , ..., n. ULTIVARIATE GENERATING FUNCTIONS 7 Proof. Recall that basing on Theorem 1 we have χ n, (cos( α ) , ..., cos( α n ) | ρ ) = X k ≥ ρ k n Y j =1 T k (cos( α j )) =12 n X i ∈{− , } ... X i n ∈{− , } (1 − ρ cos( P nk =1 i k α k ))(1 − ρ cos( P nk =1 i k α k ) + ρ ) , which is nonnegative for all α i ∈ R , i = 1 , ..., n and | ρ | < . Further R − ... R − j times ( Q ns =1 1 π √ − x s ) χ n, ( x , ..., x n | ρ ) dx ...dx j = R − ... R − ( j times Q ns =1 1 π √ − x s ) P k ≥ ρ k Q ns =1 T k ( x s ) = Q ns = j +1 1 π √ − x s . (cid:3) Let us now finish the case n = 2 . That is let us calculate χ n,m , ( x, y | ρ ) , χ n,m , ( x, y | ρ ).The case of χ n,m , ( x, y | ρ ) has been solved in e.g. [5] (Lemma 3, with q = 0) . Proposition 4. i) χ m, , ( x | ρ ) = ∞ X i =0 ρ i T i + m ( x ) = T m ( x ) − ρT m − ( x ) w ( x | ρ ) ,χ ,m , ( x | ρ ) = ∞ X i =0 ρ i U i + m ( x ) = U m − ρU m − ( x ) w ( x | ρ ) , ii) χ n,m , ( x, y | ρ ) = X k ≥ ρ k T k + n ( x ) T k + m ( y ) = (cid:0) − ρ + 2 ρ (cid:0) x + y (cid:1) − (cid:0) ρ + 3 (cid:1) ρxy (cid:1) T m ( x ) T n ( y )+ ρ (cid:0) ρ x + x − ρy (cid:1) T m ( x )( T n +1 ( y ) − T n − ( y )) + ρ (cid:0) − ρx + ρ y + y (cid:1) ( T m +1 ( x ) − T m − ( x )) T n ( y )+ ρ (cid:0) − ρ (cid:1) ( T m − ( x ) − T m +1 ( x ))( T n − ( y ) − T n +1 ( y )) /w ( x, y | ρ ) , iii) χ n,m , ( x, y | ρ ) = X j ≥ ρ j U j + n ( x ) U j + m ( y ) =( (cid:0) ρ x + x − ρy (cid:1) U m − ( x ) T n ( y )+ (cid:0) − ρ + ρ − ρx + 3 ρ xy + xy − ρy (cid:1) U n − ( y ) U m − ( x )+ T m ( x ) (cid:0) − ρx + ρ y + y (cid:1) U n − ( y ) + (cid:0) − ρ (cid:1) T m ( x ) T n ( y )) /w ( x, y | ρ ) , PAWE L J. SZAB LOWSKI iv) χ n,m , ( x, y | ρ ) = X j ≥ ρ j U m + j ( x ) T n + j ( y ) = T m ( x ) T n ( y )(1 − ρ − ρxy + 2 ρ y ) − T m ( x ) U n − ( y ) ρ ( y − x − ρy )+ U m − ( x ) T n ( y )( x − ρy )(1 + ρ − ρxy )+ U m − ( x ) U n − ( y )( y − ρ (1 − ρ + 2 ρxy − x ) /w ( x, y | ρ ) . Proof. We apply formula (3.4). For i) we take n = 1 and notice that values ofderivatives of w respect to ρ at ρ = 0 are 1 , − x, . On the way we utilize also theformula P n +1 ( x ) − xP n ( x ) = − P n − ( x ) , true for P n = U n as well as for P n = T n .To get ii) we notice that subsequent derivatives of w with respect to ρ at ρ = 0are 1 , − xy, x + 8 y − , − xy . Having this and applying directly (3.4) we getcertain defined formula expanded in powers of ρ. Now it takes Mathematica to getthis form.iii) and iv) We argue similarly getting expansions in powers of ρ. Then usingMathematica we try to get more friendly form. (cid:3) As a corollary we get formulae presented in (2.3) and (2.4) when setting n = m = 0 and remembering that T − i ( x ) = T i ( x ) , U − i ( x ) = − U i − ( x ) , for i = 0 , , Corollary 4. ∀ x, y, z ∈ [ − , , | ρ | < i) X i ≥ ρ i T i ( x ) T i ( y ) T i ( z ) = ((1 + ρ ) + 8 ρ (cid:0) x + y + z (cid:1) + 32 ρ x y z − (cid:0) ρ + 1 (cid:1) (cid:0) ρ + 3 (cid:1) ρ (cid:0) x + y + z (cid:1) + 4 (cid:0) ρ + 3 (cid:1) ρ (cid:0) x y + x z + y z (cid:1) − (cid:0) ρ + 5 (cid:1) ρ xyz (cid:0) x + y + z (cid:1) − (cid:0) ρ − ρ − ρ + 7 (cid:1) ρxyz ) /w ( x, y, z | ρ ) , ii) X i ≥ ρ i U i ( x ) U i ( y ) U i ( z ) = ((1+ ρ ) +16 ρ xyz − ρ (1+ ρ )( x + y + z )) /w ( x, y, z | ρ ) , iii) X i ≥ ρ i T i ( x ) U i ( y ) U i ( z ) = ( (cid:0) ρ + 1 (cid:1) + 8 ρ x − ρ x yz − (cid:0) ρ + 1 (cid:1) (cid:0) ρ + 3 (cid:1) ρ x +8 ρ x (cid:0) y + z (cid:1) − ρ (cid:0) − ( ρ + 2) (cid:1) xyz − (cid:0) ρ + 1 (cid:1) ρ ( y + z )) /w ( x, y, z | ρ ) , iv) X i ≥ ρ i T i ( x ) T i ( y ) U i ( z ) = ( (cid:0) ρ + 1 (cid:1) + 8 ρ (cid:0) x + y (cid:1) − (cid:0) ρ + 1 (cid:1) (cid:0) ρ + 3 (cid:1) ρ (cid:0) x + y (cid:1) +4 (cid:0) ρ + 3 (cid:1) ρ x y + 16 ρ x y z + 8 ρ z (cid:0) x + y (cid:1) − (cid:0) ρ + 2 (cid:1) ρ xyz (cid:0) x + y (cid:1) − ρ xyz − (cid:0) − ρ − ρ + 3 (cid:1) ρxyz − (cid:0) ρ + 1 (cid:1) ρ z ) /w ( x, y, z | ρ ) , where w ( x, y, z | ρ ) is given by (3.6). ULTIVARIATE GENERATING FUNCTIONS 9 Proof. Again we apply formula (3.4). Besides we take n = 3 , k = 0 for i), n = 0 , k = 3 for ii), n = 1 , k = 2 for iii) and n = 2 , k = 1 for iv). Now we have to rememberthat successive derivatives of w with respect to ρ taken at ρ = 0 are respectively1 , − xyz, − ( x + y + z ) + 4( x y + x z + y z )) , xyz (5 − x + y + z )) , − x + y + z ) + 8( x + y + z ) + 32 x y z ) , xyz (5 − x + y + z )) , − ( x + y + z ) + 4( x y + x z + y z )) , − xyz. Then we get certainformulae by applying directly formula (3.4). The expression are long and not verylegible. We applied Mathematica to get forms presented in i), ii) iii) and iv). (cid:3) Kibble–Slepian formula and related sums for Chebyshevpolynomials Let f n ( x , ..., x n | K n ) denote density of the normal distribution with zero expec-tations and covariance matrix K n such that cov( X i , X j ) = 1 for i = j, i, j = 1 , ..., n. Let ρ ij denote i, j − th entry of matrix K n . Consequently f ( x ) = exp( − x / / √ π. Let us also denote by S n a symmetric n × n matrix with zeros on the diagonal andnonnegative integers as off-diagonal entries. Let us denote i, j − th entry of thematrix S n by s ij . Recall that Kibble in 40-ties and Slepian in the 70-ties presentedthe following formula:(4.1) f n ( x , ..., x n | K n ) Q nm =1 f ( x m ) = X S ( Y ≤ i 0) = U n ( x/ 2) and [ n ] ! = 1 we see that (4.1) has beengeneralized and summed already for other polynomials. The intension of summingin [6] was to find a generalization of Normal distribution that has compact support.The attempt was partially successful since also one has obtained relatively closedform for the sum however the obtained sum was not positive for suitable values ofparameters ρ ij and all values of parameters | q | < . In the present paper we are going to present closed form of the sum (4.1) wherepolynomials H n are replaced by Chebyshev polynomials of both the first and secondkind and s ij ! are replaced by 1 . In other words we are going to find close forms for the sums: f T ( x | K n ) = X S ( Y ≤ i 1) and additionally that matrix K n + I n is positive definite. We have the following result: Theorem 2. Let us denote K n = { ( i, j ) : 1 ≤ i < j ≤ n } , β n,m = β n,m ( i n , i m ) = i n α n + i m α m . For S ⊆ K n let ρ S = Q ( n,m ) ∈ S ρ nm , b S = P ( n,m ) ∈ S β n,m , B ,...,n = B ( i , . . . , i n ) = P nj =1 i j α j . We have i) f T (cos( α ) , ..., cos( α n ) | K n ) =12 n X i ∈{− , } ... X i n ∈{− , } P nk =0 ( − k P ′ S k ⊆K n ρ S k cos( b S k ) Q nj =1 Q nm = j +1 (1 − ρ jm cos( β j,m ( i j , i m )) + ρ jm ) , ii) If n is even then f U (cos( α ) , ..., cos( α n ) | K n ) =( − n/ n Q nj =1 sin( α j ) X i ∈{− , } ... X i n + k ∈{− , } ( − P nl =1 ( i l +1) / P nk =0 ( − k P ′ S k ⊆K n ρ S k cos( B ,...,n − b S k ) Q nj =1 Q nm = j +1 (1 − ρ jm cos( β j,m ( i j , i m )) + ρ jm ) , while if n is odd then f U (cos( α ) , ..., cos( α n ) | K n ) =( − n/ n Q nj =1 sin( α j ) X i ∈{− , } ... X i n + k ∈{− , } ( − P nl =1 ( i l +1) / P n − k =0 ( − k P ′ S k ⊆K n ρ S k sin( B ,...,n − b S k ) Q nj =1 Q nm = j +1 (1 − ρ jm cos( β j,m ( i j , i m )) + ρ jm ) where S k denotes any subset of K n that contains k elements and P ′ S k ∈ K n meanssummation over all S k .Proof. Let us consider (4.2) first. Keeping in mind assertions of Proposition 3 we seethat f T (cos( α ) , ..., cos( α n ) | K n ) is the sum of 2 n summands depending on differentarrangement of values of variables i k ∈ {− , } , k = 1 , ..., n. Each summand is equalto cosine taken at P nj =1 i j s j α j . Recalling the definition of numbers s j we see thatin such sum s mj , ≤ m < j ≤ n appears twice once as s mj α m i m and s mj α j i j . Orin other words we have P nj =1 i j s j α j = P n − m =1 P nj = m +1 s mj ( α m i m + α j i j ) . Havingthis in mind we can now apply summation formula (2.11) with β = 0 and havesummed each cosine with particular system of values of the set { i j , j = 1 , ..., n } . Now it remains to sum over all such systems of values.As far as other assertions are concerned we use definition of Chebyshev poly-nomial of the second kind, formulae presented in Proposition 3. We have in thiscase P nj =1 i j ( s j + 1) α j = P nj =1 i j α j + P n − m =1 P nj = m +1 s mj ( α m i m + α j i j ) . As theresult we deal with signed sum of either sinuses or cosines depending on the factif n ( n − / s mj , ≤ m < j ≤ n ) is odd or even. Nowagain we refer to either (2.12) or (2.11) depending on parity of n ( n − / β = P nj =1 i j α j . (cid:3) Corollary 5. Both functions f T ( x | K n ) and f U ( x | K n ) are rational functions of allits arguments. Moreover they have the same denominators given by the following ULTIVARIATE GENERATING FUNCTIONS 11 formula: V n ( x | K n ) = n − Y j =1 n Y k = i +1 w ( x i , x j | ρ ij ) , where w is given by the formula (2.7).Proof. First of all notice that following formulae given in Theorem 2 the functions f T ( x | K n ) and f U ( x | K n ) are rational functions of x = cos( α ) , ..., x n = cos( α n ) . Moreover it easy to notice that all formulae have the same denominators. To findthese denominators notice that factors in each denominator referring to ( i j , i m ) and( − i j , i m ) are the same since cosine is an even function and that cosines appear solelyin denominators. Further we can group factors (1 − ρ jm cos( β j,m ( i j , i m )) + ρ jm )and (1 − ρ jm cos( β j,m ( i j , − i m )) + ρ jm ) and apply (2.8)(1 − ρ jm cos( β j,m ( i j , i m )) + ρ jm )(1 − ρ jm cos( β j,m ( i j , − i m )) + ρ jm )= w (cos α j , cos α m | ρ jm ) . since β n,m ( i n , i m ) = i n α n + i m α m . (cid:3) Corollary 6. Let us denote β kj = i k α k + i j α j , k = 1 , , j = 2 , , k < j, p = ρ ρ ρ , B , , = P j =1 i j a j ,c ( i , i , i , α , α , α , ρ , ρ , ρ ) = (1 − X ≤ k First of all notice that P k =1 P j = k +1 β kj = 2 B , , hence in particular B , , − P k =1 P j = k +1 β kj = − B , , . Then formula i) is clear basing on (2.11) with β = B , , . To get ii) notice that B , , − β = i α and B , , − β − β = − i α , similarly for other pairs (1 , 3) and (2 , B , , − P k =1 P j = k +1 β kj = − B , , . Now basing on (2.12) ii) is also clear.iii) was obtained with the help of Mathematica. (cid:3) Remark 3. With a help of Mathematica one can show for example that numer-ator of f T ( x, y, z | K ) is a polynomial of degree and consists of monomials.Numerical simulation suggest that it is a nonnegative on ( − , . Unfortunately f U ( x, y, z | K ) is not nonnegative there since we have for example f U ( − . , − . , . , | . . . . . . ) = − . . Besides notice that it happensin the case when matrix , . . . . . . is positive definite. This observation isin accordance with the general negative result presented in [6] Theorem 1. Recallthat [6] concerns something like generalization of f U to all parameters q ∈ ( − , taking into account that q -Hermite polynomials H n ( x | q ) can be identified for q = 0 with polynomials U n ( x/ . The example presented in [6] concerns the case (adoptedto q = 0) when say ρ = 0 . Hence we see that there are many sets of tuples x, y, z, ρ , ρ , ρ leading to negative values of f U . Remarks on generalization In this section firstly we are going to present q − generalization of the Chebyshevof the first kind and secondly present some remarks and observations that mighthelp to obtain formulae similar to the ones presented in Theorem 1 with Chebyshevpolynomials replaced by the so-called q − Hermite { h n } and related polynomials. q is here a certain real (in general) number such that | q | < . Since in the previouschapters we considered, so to say, the case q = 0 we will assume in this chapterthat q = 0 . To proceed further we need to recall certain notions used in q − series theory: [0] q = 0; [ n ] q = 1 + q + . . . + q n − , [ n ] q ! = Q nj =1 [ j ] q , with [0] q ! = 1 , (cid:20) nk (cid:21) q = ( [ n ] q ![ n − k ] q ![ k ] q ! , n ≥ k ≥ , otherwise . (cid:0) nk (cid:1) will denote ordinary, well known binomial coefficient.It is useful to use the so called q − Pochhammer symbol for n ≥ a ; q ) n = n − Y j =0 (cid:0) − aq j (cid:1) , ( a , a , . . . , a k ; q ) n = k Y j =1 ( a j ; q ) n . with ( a ; q ) = 1.Often ( a ; q ) n as well as ( a , a , . . . , a k ; q ) n will be abbreviated to ( a ) n and( a , a , . . . , a k ) n , if it will not cause misunderstanding.It is easy to notice that ( q ) n = (1 − q ) n [ n ] q ! and that (cid:20) nk (cid:21) q = ( ( q ) n ( q ) n − k ( q ) k , n ≥ k ≥ , otherwise . ULTIVARIATE GENERATING FUNCTIONS 13 The above mentioned formula is just an example where direct setting q = 1 issenseless however passage to the limit q −→ − makes sense.Notice that in particular [ n ] = n, [ n ] ! = n ! , (cid:2) nk (cid:3) = (cid:0) nk (cid:1) , ( a ) = 1 − a, ( a ; 1) n = (1 − a ) n and [ n ] = (cid:26) if n ≥ if n = 0 , [ n ] ! = 1 , (cid:2) nk (cid:3) = 1 , ( a ; 0) n = (cid:26) if n = 01 − a if n ≥ .i will denote imaginary unit, unless otherwise clearly stated.First we will present5.1. Generalization of the Chebyshev polynomials of the first kind. Westart with the so-called q − Hermite polynomials. There exists very large literatureon the properties as well as applications of these polynomials. Let us recall onlythat the three term recurrence satisfied by these polynomials is the following h n +1 ( x | q ) = 2 xh n ( x | q ) − (1 − q n ) h n − ( x | q ) , with h − ( x | q ) = 0 , h ( x | q ) = 1 . It is well known that the density that makes thesepolynomials orthogonal is the following f h ( x | q ) = 2 ( q ) ∞ √ − x π ∞ Y k =1 l (cid:0) x | q k (cid:1) , where l ( x | a ) = (1+ a ) − x a. Moreover characteristic function of these polynomialsare equal:(5.1) ∞ X j =0 t j ( q ) j h j ( x | q ) = 1 Q ∞ k =0 v ( x | tq k ) , where v ( x | a ) = 1 − ax + a . Recall that following [11] the density that makes q − Hermite polynomials orthogonal can be expanded in the following way: f h ( x | q ) = 2 √ − x π ∞ X k =1 ( − k − q ( k ) U k − ( x ) , where { U n } are Chebyshev polynomials of the second kind. Let us consider arelated density namely f t ( x | q ) = cπ √ − x ∞ X k =1 ( − k − q ( k ) U k − ( x ) , where constant c is to be found. To do it, let us recall that R − π √ − x U n ( x ) dx = (cid:26) if n is odd1 if n is even , hence Z − f t ( x | q ) dx = c X k ≥ ( − k − q ( k ) . Let us denote d ( q ) = P k ≥ ( − k − q ( k ) . Thus we must have c = 1 /d ( q ) . Further,as it is well known, we have: U n ( x ) U m ( x ) = min( n,m ) X j =0 U n + m − j ( x ) . (5.2) Z − f t ( x | q ) U n ( x ) dx = ( if n is odd c P k ≥ ( − k − (1 + min( n, k − q ( k ) if n is even . We need also the following transformed finite expansion, that follows formula ofpage 695 of [12] :(5.3) h n ( x | q ) = ⌊ n/ ⌋ X k =0 q k − q n − k +1 − q n − k +1 (cid:20) nk (cid:21) q U n − k ( x ) . Let us denote by γ n ( q ) the value of the integral R − f t ( x | q ) h n ( x ) dx . Hence takinginto account (5.2) and (5.3) we have γ n ( q ) = if n is odd c P ⌊ n/ ⌋ k =0 q k − q n − k +1 − q n − k +1 (cid:2) nk (cid:3) q P j ≥ ( − j − q ( j ) × (1 + min(2 j − , n − k )) if n is even . Now following Lemma 1 of [10] we deduce that polynomials t n ( x | q ) that are or-thogonal with respect to f t are of the form t n ( x | q ) = h n ( x | q ) − χ n − h n − ( x | q ) . Since we have for every t n only one parameter to determine we can do it usingcondition R − t n ( x | q ) f t ( x | q ) dx = 0 . This can be done immediately, using numbers γ n ( q ) , when n is even. For odd n we naturally get zero, since polynomials t n are thelinear combinations of odd powers of x and density f t is symmetric. To determineconstants χ n when n is odd we use the fact that we must have that Z − xt n ( x | q ) f t ( x | q ) dx = 0 , for n > , which follows directly orthogonality requirement that polynomials t n aresupposed to satisfy. But on the other hand2 xt k +1 ( x | q ) = 2 x ( h k +1 ( x | q ) − χ k − h k − ( x | q ))= h k +2 ( x | q ) + (1 − q k +1 ) h k ( x | q ) − χ k − ( h k ( x | q ) + (1 − q k − ) h k − ) . Now, knowing numbers γ k for all k ≥ χ k +1 for k ≥ . Conjectures, remarks and interesting identities. Theorem 1 suggeststhe new method of summing characteristic functions. One can formulate it in thefollowing way. ULTIVARIATE GENERATING FUNCTIONS 15 Suppose that we can guess that the form of certain multivariate characteristicfunction say for example (5.4) χ ( t ,...t n + m ) n,m ( x , . . . , x n + m | ρ, q ) = X j ≥ ρ j ( q ) j n Y k =1 h j + l k ( x k | q ) m Y i =1 t j + l n + i ( x n + i | q ) , where numbers l , . . . , l n + m are integer and | ρ | , | q | < is of the form of the ratioof two functions. Moreover, suppose that we can guess the form of the denomina-tor W n + m ( x , . . . , x n + m | ρ, q ) . Then the numerator can be obtained by the formulasimilar to (3.4) i.e. by: ∞ X j =0 ρ j j X k =0 k ! d k dρ k W n + m ( x , ..., x n + m | ρ, q ) (cid:12)(cid:12)(cid:12)(cid:12) ρ =0 × q ) j − k n Y s =1 h j − k + l s ( x s | q ) m Y v =1 t j − k + l n ++ v ( x n + v | q ) . Remark 4. There are classes of characteristic functions that have common de-nominators like for example bivariate ones described in [3] , Proposition 7 (iv) ormore generally bivariate functions of the form similar to (5.4) that were consideredby Carlitz in [1] . The point is that all these functions are at most bivariate. Thereare no results concerning more variables. Thus we have the following conjecture. Conjecture 1. Functions χ ( t ,...t n + m ) n,m ( x , . . . , x n + m | ρ, q ) for all n, m, t , . . . , t n + m are the rational functions with common denominators of the form W n,m ( x , . . . , x n + m | ρ, q ) = ∞ Y i =0 w n + m ( x , ..., x n + m | ρq i ) , where functions w n + m ( x , . . . , x n + m | ρ ) are given by iterative relationship (3.3). One-dimensional case. Now we will present a one-dimensional example, inorder to show that even in this simplest case we obtain interesting identities. Inthis example, we will so to say, derive once more formula (5.1). First of all, noticethat (1 − ae iϕ )(1 − ae − iϕ ) = 1 + a − a df = v ( x | a ) where x = cos ϕ . Moreover, wehave: W ( x | ρ, q ) = ∞ Y j =0 v ( x | ρq j ) = ( ρe iϕ ) ∞ ( ρe − iϕ ) ∞ . Let us denote indirectly function d n ( x | q ) by the relationship: n !( q ) n d n ( x | q ) = d n dρ n W ( x | ρ, q ) (cid:12)(cid:12)(cid:12) ρ =0 . We have the following lemma. Lemma 2. For | x | ≤ , | q | < , q = 0 , we have d n ( x | q ) = b n ( x | q ) , where polynomials { b n } are defined by polynomials { h n } in the following way: b n ( x | q ) = ( − n q ( n ) h n ( x | q − ) . Proof. As it is well known we have: ( ax ) ∞ = P ∞ j =0 ( − j q ( j ) a j x j ( q ) j , for | ax | ≤ | q | < . Now notice that d n da n ( xa ) ∞ (cid:12)(cid:12) a =0 = ( − n n ! x n q ( n ) ( q ) n . Hence by the Leibniz formula we have:(5.5) d n da n ( ae iϕ ) ∞ ( ae − iϕ ) ∞ (cid:12)(cid:12)(cid:12)(cid:12) a =0 = n X j (cid:18) nj (cid:19) d j da j ( e iϕ a ) ∞ (cid:12)(cid:12)(cid:12)(cid:12) a =0 d n − j da n − j ( e − iϕ a ) ∞ (cid:12)(cid:12)(cid:12)(cid:12) a =0 . Hence we have: n !( q ) n d n ( x, q ) = n X j =0 (cid:18) nj (cid:19) ( − j j ! e ijϕ q ( j )( q ) j ( − n − j ( n − j )! e − ( n − j ) iϕ q ( n − j )( q ) n − j (5.6) = ( − n n ! q ( n )( q ) n n X j =0 (cid:20) nj (cid:21) q q j ( j − n ) e i (2 j − n ) ϕ . (5.7)Now recall that (cid:2) nk (cid:3) q q k ( k − n ) = (cid:2) nk (cid:3) q − and ( q ) n /q ( n +12 ) = ( − n ( q − | q − ) , thus n !( q ) n d n ( x, q ) = n ! q n +1 ( q − | q − ) n n X j =0 (cid:20) nj (cid:21) q q j ( j − n ) e i (2 j − n ) ϕ = n ! q n +1 ( q − | q − ) n n X j =0 (cid:20) nj (cid:21) q − e i (2 j − n ) ϕ = n ! q n +1 ( q − | q − ) n h n ( x | q − ) = ( − n n ! q ( n )( q ) n h n ( x | q − ) = n !( q ) n b n ( x | q ) , where x = cos ϕ . (cid:3) Remark 5. Notice that the above mentioned (5.7) can be rewritten in the followingway: d n (cos β | ρ, q ) = ( − n q ( n ) n X j =0 (cid:20) nj (cid:21) q q j ( j − n ) e i (2 j − n ) β . Remark 6. Recall that polynomials { b n } satisfy the following three term recurrence: b n +1 ( x | q ) = − q n xb n ( x | q ) + q n − (1 − q n ) b n − ( x | q ) , with b − ( x | q ) = 0 , b ( x | q ) = 1 , as it follows from [12] (2.43). Moreover, as itfollows from [5] (3.18) after some trivial transformation polynomials { b n } satisfythe following identity: (5.8) n X j =1 (cid:20) nj (cid:21) q b n − j ( x | q ) h j + k ( x | q ) = ( if k < n ( − n q ( n ) ( q ) k ( q ) k − n h k − n ( x | q ) if k ≥ n . ULTIVARIATE GENERATING FUNCTIONS 17 Now we see that following, adapted to the present situation, formula (3.4) wehave χ t ( x | ρ, q ) = ∞ X j =0 ρ j ( q ) j h t + j ( x | q ) = 1 W ( x | ρ, q ) × ∞ X j =0 ρ j j X m =0 j − m )! ( j − m )!( q ) j − m ( q ) m b j − m ( x | q ) h m + t ( x | q )= 1 W ( x | ρ, q ) ∞ X j =0 ρ j ( q ) j j X m =0 (cid:20) jm (cid:21) q b j − m ( x | q ) h m + t ( x | q )= 1 W ( x | ρ, q ) t X j =0 (cid:20) tj (cid:21) q ( − ρ ) j q ( j ) h t − j ( x | q ) . In particular we get well-known formula for t = 0 : X j ≥ ρ j ( q ) j h j ( x | q ) = 1 W ( x | ρ, q ) , for | q | , | ρ | < | x | ≤ Two-dimensional case. Let us denote n !( q ) n d (2) n ( x, y | q ) = d n dρ n W ( x, y | ρ, q ) (cid:12)(cid:12)(cid:12) ρ =0 , where W ( x, y | ρ, q ) = Q ∞ j =0 w ( x, y | ρq j ) , where w ( x, y | a ) is defined by (2.7). Lemma 3. For θ, ϕ ∈ [0 , π ) , | q | < , we have d (2) n (cos θ, cos ϕ | q ) = ( − n q ( n ) n X m =0 (cid:20) nm (cid:21) q q m ( n − m ) (5.9) × m X j =0 (cid:20) nj (cid:21) q q j ( j − m ) e i (2 j − n )( θ + ψ ) n − m X k =0 (cid:20) n − mk (cid:21) q q k ( k − n + m ) e i (2 k − n )( θ − ψ ) . Proof. First of all notice that w (cos θ, cos ψ | ρ ) can be decomposed as w (cos θ, cos ψ | ρ ) = (1 − ρe i ( θ + ψ ) )(1 − ρe − i ( θ + ψ ) )(1 − ρe i ( θ − ψ ) )(1 − ρe − i ( θ − ψ ) ) , hence W (cos θ, cos ψ | ρ, q ) = ( ρe i ( ϕ + θ ) ) ∞ ( ρe − i ( ϕ + θ ) ) ∞ ( ρe i ( ϕ − θ ) ) ∞ ( ρe − i ( ϕ − θ ) ) ∞ = W (cos( θ + ϕ ) | ρ, q ) W (cos( θ − ϕ ) | ρ, q ) . Now using (5.7) we see that d n dρ n W (cos β | ρ, q ) (cid:12)(cid:12)(cid:12) ρ =0 = ( − n n ! q ( n ) ( q ) n P nj =0 (cid:2) nj (cid:3) q q j ( j − n ) e i (2 j − n ) β , so n !( q ) n d (2) n (cos θ. cos ψ | q ) = n X m =0 (cid:18) nm (cid:19) (( − m m ! q ( m )( q ) m m X j =0 (cid:20) mj (cid:21) q q j ( j − m ) e i (2 j − n )( θ + ψ ) )( − n − m × ( n − m )! q ( n − m )( q ) n − m n − m X k =0 (cid:20) n − mk (cid:21) q q k ( k − n + m ) e i (2 k − n )( θ − ψ ) = ( − n n !( q ) n q ( n ) n X m =0 (cid:20) nm (cid:21) q q m ( n − m ) m X j =0 (cid:20) mj (cid:21) q q j ( j − m ) e i (2 j − n )( θ + ψ ) × n − m X k =0 (cid:20) n − mk (cid:21) q q k ( k − n + m ) e i (2 k − n )( θ − ψ ) . (cid:3) Using Mathematica we can find d (2) n ( x, y | q ) for n = 1 , , x = cos θ, and y = cos ψ getting: d (2)1 ( x, y | q ) = − b ( x | q ) b ( y | q ) , d (2)2 ( x, y, | q ) = q − ( b ( x | q ) b ( y | q ) − (1 − q )) ,d (2)3 ( x, y | q ) = − q − ( b ( x | q ) b ( y | q ) − q ( q ) ( q ) b ( x | q ) b ( y | q )) ,d (2)4 ( x, y, | q ) = q − ( b ( x | q ) b ( y | q ) − q ( q ) ( q ) ( q ) b ( x | q ) b ( y | q ) + q ( q ) ( q ) ) . This allows us to venture the Conjecture. Conjecture 2. d (2) n can be expressed by polynomials b n in the following way: d (2) n ( x, y | q ) = ( − n q − ( n ) ⌊ n/ ⌋ X j =0 β n,j ( q ) b n − j ( x | q ) b n − j ( y | q ) , where β n,j ( q ) are some functions of q only and moreover β n, = 1 . Now knowing that ∞ X j =0 ρ j ( q ) j h n ( x | q ) h n ( y | q ) = ( ρ ) ∞ W ( x, y | ρ, q ) , we get the following identity: ∞ X k =0 ρ j ( q ) j j X m =0 (cid:20) jm (cid:21) q d (2) m ( x, y | q ) h j − m ( x | q ) h j − m ( y | q ) = ∞ X k =0 ( − k q ( k ) ρ k ( q ) k , for all | x | , | y | ≤ , | ρ | , | q | < , which can be easily generalized following modificationof formula given in assertion i) of Lemma 3 in [4] to: ∞ X k =0 ρ j ( q ) j j X m =0 (cid:20) jm (cid:21) d (2) m ( x, y | q ) h j − m + t ( x | q ) h j − m + s ( y | q )= V t,s ( x, y | ρ, q ) ∞ X k =0 ( − k q ( k ) ρ k ( q ) k , ULTIVARIATE GENERATING FUNCTIONS 19 for t, s ∈ N ∪{ } , where V t,s ( x, y | ρ, q ) denotes certain polynomial of the order t + s in x and y. Proofs Proof of Proposition 3. Proof is by induction. For n = 1 and k = 1 we have in caseof (2.9) cos α = (cos α + cos( − α )) while in case of (2.10) we getsin α cos α = − 14 (sin( − α − α ) + sin( − α + α ) − sin( α − α ) − sin ( α + α ))= 12 (sin( α + α ) + sin( α − α )) . Hence assume that they are true for n = m. In the case of the first one we have m +1 Y j =1 cos( ξ j ) = cos( ξ m +1 ) m Y j =1 cos( ξ j ) =12 m X i ∈{− , } ... X i m ∈{− , } cos( m X k =1 i k ξ k ) cos( ξ m +1 )= 12 m +1 × X i ∈{− , } ... X i m ∈{− , } (cos( m X k =1 i k ξ k + ξ m +1 ) + cos( m X k =1 i k ξ k − ξ m +1 )) . On the way we used the fact that cos( α ) cos( β ) = (cos( α − β ) + cos( α + β )) / . In the case of the second one we first consider the case of k = 0. Assuming that m is even we get: m +1 Y j =1 sin( ξ j ) = sin( ξ m +1 ) m Y j =1 sin( ξ j ) = ( − m/ m × X i ∈{− , } ... X i m ∈{− , } ( − P mk =1 ( i k +1) / cos( m X k =1 i k ξ k ) sin( ξ m +1 )= ( − m/ m +1 X i ∈{− , } ... X i m ∈{− , } ( − P mk =1 ( i k +1) / × (sin( m X k =1 i k ξ k + ξ m +1 ) − sin( m X k =1 i k ξ k − ξ m +1 )) = − ( − m/ m +1 X i m +1 ∈{− } X i ∈{− , } ... X i m ∈{− , } ( − P m +1 k =1 ( i k +1) / sin( m +1 X k =1 i k ξ k ) − ( − m/ m +1 X i m +1 ∈{− } X i ∈{− , } ... X i m ∈{− , } ( − P m +1 k =1 ( i k +1) / sin( m +1 X k =1 i k ξ k ) . We used the fact that sin( α ) cos( β ) = (sin( α − β ) + sin( α + β )) / . The case of m odd is treated in the similar way.Now to consider general case we expand both products of sines and cosines andnotice that because of the properties of products of cosines with sines or cosines presented above, the number of sign changes does not depend on variables ξ , ..., ξ k but only the number of negative and positive expressions doubles. (cid:3) Proof of Lemma 1. (2.11) We putcos( β + n X i =1 k i α i ) = exp( iβ + n X j =1 ik j α j ) / − iβ − n X j =1 ik j α j ) / . So X k ≥ ... X k n ≥ ( n Y i =1 ρ k i i ) exp( iβ + n X j =1 ik j α j ) / iβ ) n Y j =1 − ρ j exp( iα j ) . Similarly: X k ≥ ... X k n ≥ ( n Y i =1 ρ k i i ) exp( − iβ − n X j =1 ik j α j ) / − iβ ) n Y j =1 − ρ j exp( − iα j ) . Thus X k ≥ ... X k n ≥ ( n Y i =1 ρ k i i ) cos( β + n X j =1 ik j α j )= exp( iβ ) Q nj =1 (1 − ρ j exp( − iα j )) + exp( − iβ ) Q nj =1 (1 − ρ j exp( iα j ))2 Q nj =1 (1 + ρ j − ρ j cos( α j )) . Now notice that exp( − iβ ) n Y j =1 (1 − ρ j exp( iα j )) = n X j =1 ( − j X M j,n ⊆ S n Y k ∈ M j,n ρ k exp( − iβ + i X k ∈ M j,n α k ) . To get (2.12) we putsin( β + n X j =1 k j α j ) = exp( iβ + n X j =1 ik j α j ) / i − exp( − iβ − n X j =1 ik j α j ) / i. So X k ≥ ... X k n ≥ ( n Y j =1 ρ k i j ) exp( iβ + i n X j =1 k j α j ) / i = exp( iβ ) 12 i n Y j =1 − ρ j exp( iα j ) . Similarly X k ≥ ... X k n ≥ ( n Y i =1 ρ k i i ) exp( − iβ − i n X i =1 k i α i ) / i = exp( − iβ ) 12 i n Y j =1 − ρ j exp( − iα j ) . So X k ≥ ... X k n ≥ ( n Y i =1 ρ k i i ) sin( β + n X i =1 k i α i ) =12 i exp( iβ ) Q nj =1 (1 − ρ j exp( − iα j )) − exp( − iβ ) Q nj =1 (1 − ρ j exp( iα j )) Q nj =1 (1 + ρ j − ρ j cos( α j )) . ULTIVARIATE GENERATING FUNCTIONS 21 (cid:3) Proof of Theorem 1. Proof is based on the following observation. For α s ∈ R , t s ∈ Z , s = 1 , ..., n + k , | ρ | < n is odd then, X j ≥ ρ j n Y s =1 U j + t s (cos( α s )) n + k Y s = n +1 T j + t s (cos( α s )) =(6.1) ( − ( n +1) / n + k Q ni =1 sin( α i ) X i ∈{− , } ... X i n + k ∈{− , } ( − P nk =1 ( i k +1) / × (sin( P ns =1 i s ( t s + 1) α s + P n + ks = n +1 i s t s α s ) − ρ sin( P ns =1 i s t s α s + P n + ks = n +1 i s ( t s − α s ))(1 − ρ cos( P n + ks =1 i s α s ) + ρ ) , while when n is even or zero we get: X j ≥ ρ j n Y s =1 U j + t s (cos( α s )) n + k Y s = n +1 T j + t s (cos( α s )) =(6.2) ( − n/ n + k Q ni =1 sin( α i ) X i ∈{− , } ... X i n + k ∈{− , } ( − P nk =1 ( i k +1) / × cos( P ns =1 i s ( t s + 1) α s + P n + ks = n +1 i s t s α s ) − ρ cos( P ns =1 i s t s α s + P n + ks = n +1 i s ( t s − α s )(1 − ρ cos( P n + ks =1 i s α s ) + ρ ) . To justify it we use (2.1) first, then basing on Proposition 3 we deduce that for n odd we have to sum sinuses of the following arguments: n X s =1 l s (( j + 1) α s + t s α s ) + n + k X s = n +1 l s ( jα s + t s α s )= j n + k X s =1 l s α s + n X s =1 l s ( t s + 1) α s + n + k X s = n +1 l s t s α s . We identify ” α ” and ” β ” and apply formula (2.13). When n is even or zero weargue in a similar way but this time with cosines.Now let us analyze polynomial w n . Notice that denominator in both (6.1) and(6.2) is of the form w k + n (cos( α ) , ..., cos( α k + n ) | ρ ) =(6.3) Y i ∈{− , } ... Y i k + n ∈{− , } (1 − ρ cos( n + k X s =1 i s α s ) + ρ ) . To get (6.3) we will argue by induction. Let us replace n + k by m to avoid confusion.To start with m = 1 for m = 2 we recall (2.8). Hence (6.3) is true for m = 1 , . Let us hence assume that formula is true for n = k + 1 . Hence taking α = α k +1 and β = P ks =1 i s α s and noticing that i k = 1we get: w k +1 (cos( α ) , ..., cos( α k +1 ) | ρ ) = Y i ∈{− , } ... Y i k ∈{− , } ((1 − ρ cos( k − X s =1 i s α s + i k ( α k − i k α k +1 )) + ρ ) × (1 − ρ cos( k − X s =1 i s α s + i k ( α k + i k α k +1 )) + ρ ))= w k (cos( α ) , ..., cos( α k + α k +1 ) | ρ ) w k (cos( α ) , ..., cos( α k − α k +1 ) | ρ ) . by induction assumption. Now it is elementary to see that polynomials w n satisfyrelationship (3.3). Similarly the remarks concerning order of symmetry and orderof polynomials w n follow directly (6.3).Now let us multiply both sides of (6.1) and (6.2) by w n + k ( x , ..., x n + k | ρ ). Wesee that this product is equal to right hand sides of these equalities with an obviousreplacement cos( α s ) − > x s , s = 1 , ..., n + k. Inspecting (6.1) and (6.2) we noticethat these right hand sides are polynomials of order 2(2 n + k − − 1) + 1 = 2 n + k − ρ. Thus these polynomials can be regained by using well known formula p n ( x ) = n X i =0 x n a n = n X j =0 x j j ! d j dx j p n ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x =0 . This leads directly to the differentiation of products of w n + k ( x , ..., x n + k | ρ ) andright hand side of (1.1). Now we apply Leibniz formula: d n dx n [ f ( x ) g ( x )] (cid:12)(cid:12)(cid:12)(cid:12) x =0 = n X j =0 (cid:18) ni (cid:19) d j dx j f ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x =0 d n − j dx n − j g ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x =0 . and notice that d k dρ k X j ≥ ρ j n Y s =1 T j + t s ( x s ) n + k Y s =1+ n U j + t s ( x s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ =0 = k ! n Y s =1 T k + t s ( x s ) n + k Y s =1+ n U k + t s ( x s ) . Having this we get directly (3.1). (cid:3) References [1] Carlitz, L. Generating functions for certain q -orthogonal polynomials. Collect. Math. (1972), 91–104. MR0316773 (47 Adv. in Appl. Math. (2016), 70–92.[3] Szab lowski, Pawe l J. Multidimensional $q$-normal and related distributions—Markov case.Electron. J. Probab. 15 (2010), no. , 1296–1318. MR2678392[4] Szab lowski, Pawe l J. On the structure and probabilistic interpretation of Askey-Wilson den-sities and polynomials with complex parameters. J. Funct. Anal. 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On affinity relating two positive measures and the connection coefficientsbetween polynomials orthogonalized by these measures. Appl. Math. Comput. (2013),no. 12, 6768–6776. MR3027843[11] Szab lowski, Pawe l J. (2009) q − Gaussian Distributions: Simplifications and Simulations, Jour-nal of Probability and Statistics , 2009 (article ID 752430)[12] Szab lowski, Pawe l, J. On the q − Hermite polynomials and their relationship withsome other families of orthogonal polynomials, Dem. Math. Emeritus in Department of Mathematics and Information Sciences,, Warsaw Univer-sity of Technology, ul Koszykowa 75, 00-662 Warsaw, Poland E-mail address ::