## Abstract

Identities of the formare proved. Here *W*_{1} is either of the Whittaker functions *W*_{κ,μ} or *M*_{κ,μ} and *W*_{2} is either of or . The function *g* has, piecewise, a form that involves a hypergeometric function of a rational function of *z* and *ζ*. These identities make possible the calculation of explicit global propagators for certain singular hyperbolic equations and degenerate hyperbolic equations in two variables of the form

## 1. Introduction

Singular hyperbolic operators in two variables of the form(1.1)and degenerate hyperbolic operators of the form(1.2)exhibit interesting features that vary qualitatively with the real parameter *λ* and with the parity of the integer *k*>1 (Treves 1974; Tanaguchi & Tozaki 1980). While studying these operators, it is natural to take advantage of translation invariance in one direction by taking a partial Fourier–Laplace transform. This leads to the ordinary differential operators(1.3)where *s* is the transform parameter. Taking *z*=2*sx*^{k}/*k* or *z*=2*st*^{k}/*k* and conjugating by *x*^{c/2} or *t*^{c/2}, respectively, where *c*=1−(1/*k*), converts (1.3) to the Whittaker operator(1.4)

It follows from these considerations that the propagators for the operators (1.1) and (1.2) are inverse Laplace transforms of certain multiples of Green's functions for the Whittaker operator or the associated Kummer operator. These Green's functions, in turn, are themselves products of Whittaker or Kummer functions.

To calculate the (global) propagators explicitly in this way requires explicit formulae for inverse transforms of products of various pairs of Whittaker functions. One such formula (corollary to theorem 3.1) is known (Erdélyi *et al*. 1954), and was used in Greiner & Stein (1978) to calculate the Green's function for some cases of □_{b}, a subelliptic operator that occurs in the theory of boundary problems in several complex variables. We have found no proof or reference for this formula in the literature. In this paper, we prove it and the remaining formulae that arise in calculating the propagators for (1.1) and (1.2).

Notation and basic identities for Kummer and Whittaker functions are reviewed in §2. The formulae that exhibit products as inverse Laplace transforms are stated in §3 and proved in §4.

The complete formulae for the hyperbolic propagators and an analysis of the behaviour in the parameter *λ* are the subjects of a separate paper (Beals & Kannai in preparation). We note that in each case the propagator is, piecewise, a triple product. Two factors are powers of rational functions of *x* and *t*, and the third is the composition of a hypergeometric function with a rational function of *x* and *t*.

This paper and Beals & Kannai (in preparation) have some (minimal) overlap with Bentrad (2006), where solutions ofare given as series involving hypergeometric functions.

The change of variables *y*=*x*^{k}/*k* in (1.1) or *y*=*t*^{k}/*k* in (1.2), together with a further linear change of coordinates, leads to the Euler–Poisson–Darboux (EPD) operatorwith parameters that depend on both *k* and *λ*. However, most results on EPD are only semi-global (Darboux 1889; Copson 1958, 1975; Friedlander & Heins 1968), limited to a region like *w*_{0}<*r*+*s*<*w*_{1} in which the qualitative variation with *λ* (and with the parity of *k*) does not appear.

The formulae proved in this paper are more general than are needed for the application to (1.1) and (1.2), in that the second index *κ*′ needed for the application is always either *κ*′=*κ* or *κ*′=−*κ*.

## 2. Kummer and Whittaker functions

We use the notation of Abramowitz & Stegun (1965), though for later convenience we write *c* in place of *b* for the second parameter in Kummer functions. The following formulae are found in §§13.1 and 13.2 of this reference. Kummer's equationhas, for *c*≠0, −1, −2, …, an entire solution(2.1)which, for has the integral representation(2.2)There is a solution that is bounded as *z*→+∞ given byFor , *U* has an integral representation(2.3)

Now, set(2.4)Then Whittaker's operator (1.4) has solutions(2.5)A solution that decays exponentially as *z*→+∞ is(2.6)For appropriate ranges of the parameters, we can deduce integral representations for *M*_{κ,μ} and *W*_{κ,μ} from (2.2) and (2.3).

## 3. Statement of results

We state three identities that are valid for the given values of the parameters *μ*, *κ* and *κ*′ without further qualification. Suitably interpreted (e.g. in the sense of distributions), they may be extended by analytic continuation to generic values of these parameters.

We use the standard notation *F*(*α*, *β*, *γ*; *x*) for the Gauss hypergeometric function

*Suppose that* *and* . *Then*(3.1)*where*(3.2)

The two forms of *f* given in (3.2) are related by Pfaff's identity(3.3)The same is true for the two forms of *f* on various intervals in the statements of theorems 3.4 and 3.5.

Setting *τ*=*pt*, *z*=*αp* and *ζ*=*βp* converts the identities (3.1) and (3.2) to the following form, which is the formula given in §4.21 (8), p. 213 of Erdélyi *et al*. (1954).

*Under the assumptions of* *theorem 3.1**, for α, β, p*>0,(3.4)

For each of theorems 3.4 and 3.5 and for 0<*β*<*α*, there is a corresponding formula that expresses a product of Whittaker functions of variables *αp* and *βp*, multiplied by , as the Laplace transform of an expression that is, piecewise, similar to the last line of (3.4). In theorem 3.4 there is one such expression on the interval 0<*t*<*β* and a second on the interval *β*<*t*<∞. In theorem 3.5, there are three such expressions, corresponding to the intervals 0<*t*<*β*, *β*<*t*<*α* and *α*<*t*<*α*+*β*.

*Suppose that* *and* . *Then*(3.5)*where*(3.6)

(3.7)

*Suppose that* , *and* 0<*ζ*<*z*. *Then*(3.8)*where*(3.9)

(3.10)

(3.11)

The following lemma is key.

*Given complex α, β, γ with* *, set*(3.12)*for* , . *Then*(3.13)

It is enough to prove this when *x*, *y*>0. The idea is that under a Möbius transformation that fixes 0 and 1 and moves the singularity 1/*y* to ∞, the integral simplifies to a standard integral representation of the hypergeometric function. In fact, given *ρ*>0, the change of variablesmaps the interval [0,1] to itself andSince the exponents sum to 0, the powers of *s*+*ρ* from the denominators cancel andTake 1+*ρ*=1/*y*, so that the last factor becomes 1. Then the integral becomesUp to a numerical factor, this integral is Euler's integral formand we obtain the first identity in (3.13). Choosing 1+*ρ*=1/*x* leads to the second identity in (3.13). ▪

The two identities in (3.13) are related by Pfaff's identity (3.3).

## 4. Proofs of theorems 3.1, 3.4 and 3.5

To lighten the notational burden we setNote that 1+2*μ*=*a*+*b*, 1−2*μ*=*a*′+*b*′ and *a*+*b*+*a*′+*b*′−2=0.

Using the identity and the identities (2.3) and (2.6) we obtain(4.1)Set *τ*=*zτ*_{1}+*ζτ*_{2} and *u*=*zτ*_{1}, so that *τ*_{1}=*u*/*z*, *τ*_{2}=(*τ*−*u*)/ζ and 0≤*u*≤*τ*. Change variables again to *u*=*στ*, 0≤*σ*≤1. The double integral in (4.1) is(4.2)whereThe last integral is (3.12) with *x*=−*τ*/*z*, *y*=*τ*/(*τ*+*ζ*), *α*=*a*, *β*=1−*b*, *γ*=*a*+*a*′, so(4.3)The identities (4.1)–(4.3) give (3.1) and (3.2). ▪

Using the identity and the identities (2.2), (2.3), (2.5) and (2.6), we obtain(4.4)Set , , so thatThen the integral in (4.4) iswhere

When *τ*<*ζ* in (4.4) we set *u*=*στ* to getThis last integral is (3.12) with *x*=*τ*/(*z*+*τ*), *y*=*τ*/*ζ*, *α*=1−*a*′, *β*=*a*, *γ*=*a*+*b*′, so(4.5)The identities (4.3) and (4.5) give (3.6).

When *τ*>*ζ* we set *u*=*σζ* so thatThis last integral is (3.12) with *x*=*ζ*/(*z*+*τ*), *y*=*ζ*/*τ*, *α*=1−*a*′, *β*=*a*, *γ*=*a*+*b*, so(4.6)The identities (4.3) and (4.6) give (3.7). ▪

The integrals leading to (4.5) and (4.6) are Appell hypergeometric functions of two variables (Erdélyi *et al*. 1953, p. 230, §5.8.1, (1)). The reduction formula (Erdélyi *et al*. 1953, p. 238, §5.10 (1)) and (4.3) can then be used to deduce the first form of (4.5) and the second form of (4.6).

Using the identities (2.2) and (2.5) we obtainSet *τ*_{j}=1−*σ*_{j}, so that this becomes(4.7)Let *τ*=*σ*_{1}*z*+*σ*_{2}*ζ* and *u*=*σ*_{1}*z*, so that 0≤*τ*≤*z*+*ζ* andThen the double integral in (4.7) becomeswhere(4.8)

For 0<*τ*<*ζ*, the interval of integration in (4.8) is 0<*u*<*τ*, so we take *u*=*τσ* and obtainThe last integral is (3.12) with *x*=*τ*/*z* and *y*=−*τ*/(*ζ*−*τ*), *α*=*b*, *β*=1−*a*, *γ*=*b*+*b*′, so(4.9)The identities (4.7)–(4.9) give (3.9).

For *ζ*<*τ*<*z*, the interval of integration in (4.8) is *τ*−*ζ*<*u*<*τ*, so we take *u*=*τ*−*ζ*+*ζσ*. Let *w*=*z*+*ζ*−*τ*. ThenThe last integral is (3.12) with *x*=*ζ*/*w*, *y*=−*ζ*/(*τ*−*ζ*), *α*=*a*′, *β*=1−*a*, *γ*=*a*′+*b*′, so(4.10)The identities (4.7), (4.8) and (4.10) give (3.10).

For *z*<*τ*<*z*+*ζ*, the interval of integration in (4.8) is *τ*−*ζ*<*u*<*z*, so we take *u*=*τ*−*ζ*+*σw* with *w*=*z*+ζ−*τ*. ThenThe last integral is (3.12) with *x*=*w*/*ζ*, *y*=−*w*/(*τ*−*ζ*), *α*=*a*′, *β*=1−*b*′, *c*=*a*+*a*′, so(4.11)The identities (4.7), (4.8) and (4.11) give (3.11). ▪

## Acknowledgments

Y.K. acknowledges a useful discussion of these topics with Jeremy Schiff. Both authors thank the referees for useful comments.

## Footnotes

- Received October 3, 2007.
- Accepted December 5, 2007.

- © 2008 The Royal Society