ISSN 385-75 Working papers of the Department of Economics University of Perugia (IT) The Heat Kernel on Homogeneous Trees and the Hypergeometric Function Hacen Dib Mauro Pagliacci Working paper No 6 December 04
THE HEAT KERNEL ON HOMOGENEOUS TREES AND THE HYPERGEOMETRIC FUNCTION HACEN DIB AND MAURO PAGLIACCI Abstract The combinatorial coefficients in the explicit solution of the heat equation on homogeneous trees are expressed by a hypergeometric function The Heat Equation on Homogeneous Trees Let T be a homogeneous tree of degree q +, ie a tree such that every vertex belongs exactly to q + edges If u and v are vertices of the tree, the natural distance between u and v is the number of edges in the path from u to v A function defined on T is harmonic if its value at every vertex is the average of the value at its neighbors We consider the nearest neighbor random walk on T whose transition probabilities are p(u, v) if d(u, v) = and zero otherwise The random walk is isotropic if d(u, v) = Let v q+ n be the vertex visited by the random walk on T after n steps and let p (n) (u, v) = Pr[v n = v, v 0 = u] the probability to go from u to v in n steps In other words p (n) (u, v) are the elements of the n-th power of the transition matrix with elements p(u, v) The Green kernel G(u, v) = n=0 p(n) (u, v) represents the expected number of visits to v, starting from u To introduce the heat equation on T, we consider functions f : T N C, Denoting by f(v, k) = f k (v), v is the space variable and k the time variable The Laplacian of the function f is defined by L T f k (v) = p(u, v)f k (u) f k (v), u v where u v means that d(u, v) = Analogously to the classical case, we can consider the heat equation on T the following boundary value problem: L T f k (v) = f k+ (v) f k (v), given f 0 (v) () 99 Mathematics Subject Classification Primary 60J60, 05C05, 33C0 Key words and phrases Homogeneous trees, transition probabilities, heat semigroup, hypergeometric functions
HACEN DIB AND MAURO PAGLIACCI The heat equation on T was solved explicitly in [3] in the isotropic case using a combinatorial approach and in [] using the inversion formula for the Radon transform The heat equation can be considered also as a convolution problem In particular, if e is a reference vertex in T (ie the root of T ), we can consider the probability measure on T given by µ (v) = p(e, v), if v = (where v = d(e, v)) and µ (v) = 0 otherwise In the isotropic case, we have µ (v) = if v = and µ q+ (v) = 0 otherwise Then the heat equation can be written in the following way: f k (v) = u v p(v, u)f k (u), given f 0 (v), and therefore: f k (v) = f k µ (v) given f 0 (v) In [3] is proved (Theorem ) that the solution of the heat equation is given by where [k/] f k (v) = d q (k, k j) f 0 (u), j=0 d(v,u)=k j d q (k, k j) = ( ) k j i=0 ( k j + i k j ) k j i k j + i ( ) i The principal aim of the present paper is to find a new form for the explicit solution of the heat equation using the special functions This fact can be used to find an asymptotic estimate for the heat kernel The hypergeometric function 3 F Il this section we recall the definition of the hypergeometric function 3F For more details see [] This special function can be used to write the heat kernel described in section and can be used also to give an asymptotical estimate of the heat kernel The Hypergeometric function 3 F is defined by 3F (α, β, γ; µ, ν; y) = k=0 (α) k (β) k (γ) k y k, (µ) k (ν) k k!
HEAT KERNEL ON TREES AND HYPERGEOMETRIC FUNCTION 3 where, for z C, (z) l = z(z ) (z + l ) = Γ(z+l), with Γ the Γ(z) Euler Gamma function The series converges for y > and is well defined for µ, ν Z The main idea of this paper is to use the function 3 F to write the heat kernel on the homogeneous trees: it can be useful to obtain the an asymptotic estimate of the heat kernel Proposition The coefficients d q (k, k j) is section can be written as: j k k j + d q (k, k j) = () k j + [ 3 F j, k j, + Proof We have that ( ) k j ((k j)(k j + ) ; k j +, k j (k j)(k j + ) ; k j ] d q (k, k j) = ( )j k We can write: ( k j + i i=0 i ) k j i k j + i ( ) i q A k,j (t) = ( ) k j + i k j i c i t i, with c i = i k j + i i=0 We can express A k,j as a polynomial of t instead of t We remark that : ( ) ( ) k j + i k j + i c i =, with i j i i Therefore: where A k,j (t) = B k,j (t) tb k,j (t), B k,j (t) = Now, using the identity: ( ) k j + i t i i i= α! ( d dt )α t β+α = ( β + α α ) t β, (α, β N),
4 HACEN DIB AND MAURO PAGLIACCI we have: ( ) k j ( ) d t k t k j B k,j (t) = (k j )! dt t After a suitable change of variable we can write: Therefore: Then: B k,j (t + ) = A k,j (t + ) = k! (k j )! k! (k j )! k! (t + ) (k j)! i t l!(k j + l) l!(j l) t l l!(j l)!(k j + l) j t l l!(j l)!(k j + + l) with A k,j (t + ) = a l t t, Since a l a l k!(k j + )(l + (k j)(k j+) ) k j a l = l!(j l)!(k j)!(k j + l)(k j + l + ) is rational in l, we have: a l = ( ) l k j + k j + ( k j ) ( j)l (k j) l ( + (k j)(k j+) k j (k j + ) l ( (k j)(k j+) k j ) l This is the coefficient of the Hypergeometric function 3 F The 3 F series is finite because ( j) l = 0, if l j + Corollary The coefficients d q (k, m) in the solution of the heat equations on homogeneous trees can be written as: d q (k, m) = () k+m 3 F [ m k (m + ) k + m +, m + k, + Γ(k + ) Γ( k m+ )Γ( k+m+ (m + )(m + k) (m ) ) ) l ; m + k + 4, (m + )(m + k ; (m ) ]
HEAT KERNEL ON TREES AND HYPERGEOMETRIC FUNCTION 5 Proof By the change of variables m = k m and remarking that ( ) k k! = j j!(k j)! = Γ(k ) Γ(j + )Γ(k j + ) References [] JM Cohen, M Pagliacci, Explicit Solution for the Wave Equation on Homogeneous trees, Adv Appl Math 5 (994), 390 403 [] NN Lebedev, Special Functions and their Applications, Dover Pubblications, Inc,, New York, 07 [3] M Pagliacci, Heat and wave Equations on Homogeneous Trees, Boll Un Mat It (7) Sez A 7 (993), 37 45 Département de Mathématiques, Université de Tlemcen, Tlemcen 3000 Algerie, Dipartimento di Economia, Università di Perugia, Via A Pascoli, 063 Perugia, Italy