Nicholson-Type Integral Relations

• Nicholson-type integral relations are integral identities that convert products of special functions into single convergent integrals with hyperbolic kernels.
• They bridge classical methods like Bessel and Airy integrals with modern transforms in Heun, Jacobi, and Laguerre functions, enhancing analytic continuation and numerical stability.
• These relations also extend to boundary-value and Hilbert-transform frameworks, clarifying links with delay differential equations and broader kernel theories.

Nicholson-type integral relations are integral identities in which a product, boundary-value pairing, or solution transform for a second-order differential equation is rewritten as a one-dimensional integral with a structured kernel. In the classical special-function setting, the defining pattern is that a product of two solutions at the same argument is represented by an integral over a hyperbolic parameter; in broader kernel theory, the same label is often used structurally for transforms of the form “solution equals integral of kernel times solution,” with the kernel satisfying a companion differential equation. Modern work places this pattern in several distinct but connected settings: the classical Airy–Bessel relation attributed to Nicholson, hyperbolic product formulas for parabolic cylinder and Jacobi-type functions, kernel transforms for Heun-family equations, and Nicholson-adjacent Hilbert-transform identities for Bessel functions. The term also appears in delay-differential models of population dynamics, where it denotes a different lineage associated with Nicholson’s blowflies rather than classical special-function product formulas (Tabrizi et al., 2016, Nasri, 2015, El-Jaick et al., 2013, Ossandón et al., 2020).

1. Structural definition and characteristic form

In the narrow special-function sense, a Nicholson-type relation has the canonical pattern

$$\text{product of two special functions at the same argument} = \int_0^\infty \text{kernel depending on a hyperbolic parameter}\,dt,$$

so that the dependence on the integration variable enters through functions such as $\cosh t$ and $\sinh t$. The data on parabolic cylinder functions states explicitly that this is why the resulting formulas are called Nicholson-type: the designation is structural, not merely terminological, and is modeled on the classical Bessel formula

$$J_\nu^2(x)+Y_\nu^2(x)=\frac{8}{\pi^2}\int_0^\infty K_0(2x\sinh t)\cosh(2\nu t)\,dt.$$

In that usage, the main analytic content is the conversion of a product into a single absolutely convergent integral with a unified kernel (Nasri, 2015).

A second, broader usage appears in Heun theory. There the central object is an integral transform

$\mathcal H(x)=\int K(x,y)H(y)\,dy$

such that the transform again solves the same differential equation, provided the kernel satisfies a companion equation such as $(M_x-M_y)G=0$ and the associated bilinear concomitant vanishes at the endpoints. This is not a classical Nicholson product formula, but it is closely analogous in mechanism: a bilocal kernel intertwines solutions of the same second-order ODE, and the boundary term plays the role of the structural consistency condition (El-Jaick et al., 2010, El-Jaick et al., 2013).

A recurring misconception is that every occurrence of “Nicholson-type” refers to special-function product integrals. In population dynamics the phrase instead labels delayed Nicholson systems. One paper studies an $\omega$-periodic two-patch Nicholson-type delay system with nonlinear density-dependent mortality and uses one-period averaged integral balances and Brouwer degree, not Green-function product formulas or hyperbolic kernels. Its “Nicholson-type integral relations” are therefore period-averaged balance identities of a different kind (Ossandón et al., 2020).

2. Classical Airy–Bessel relation and Nicholson’s original method

A central historical prototype is the relation between the oscillatory Airy integral and modified Bessel functions revisited in "On the relation between Airy integral and Bessel functions revisited" (Tabrizi et al., 2016). The paper considers

$$\int_{0}^{\infty}\cos\!\left[\frac{3}{2}\xi\!\left(x+\frac{x^{3}}{3}\right)\right]\,dx$$

and reduces it, by the substitution $\frac12\xi x^3=\omega^3$, to the Airy integral

$$A(\rho)=\int_{0}^{\infty}\cos(\omega^{3}+\rho\omega)\,d\omega, \qquad \rho=3\left(\frac{\xi}{2}\right)^{2/3}.$$

The resulting differential equation

$\cosh t$0

is converted to Bessel normal form, and matching coefficients yields the characteristic order $\cosh t$1 (Tabrizi et al., 2016).

The principal Nicholson-type identities in that treatment are

$\cosh t$2

and the derivative companion

$\cosh t$3

The orders $\cosh t$4 and $\cosh t$5 arise in different ways: $\cosh t$6 comes from the reduction of the Airy ODE to the Bessel normal form, whereas $\cosh t$7 is generated only after differentiating the $\cosh t$8-formula and using the standard recurrences for $\cosh t$9 (Tabrizi et al., 2016).

That paper is also notable for separating two proof styles. Nicholson’s original route derives an ODE for the oscillatory integral and identifies it with a Bessel equation; the alternative proof uses the Bowman transformation

$\sinh t$0

with a purely imaginary $\sinh t$1, so that the Bessel argument becomes imaginary and the solution is rewritten in terms of $\sinh t$2 and then $\sinh t$3. The paper explicitly treats these formulas as standard relations used in radiation physics (Tabrizi et al., 2016).

3. Parabolic cylinder products and confluent-hypergeometric kernels

A major modern extension is the representation of products of parabolic cylinder functions in terms of Laplace and Fourier transforms of Kummer confluent hypergeometric functions in "Product of parabolic cylinder functions involving Laplace transforms of confluent hypergeometric functions" (Nasri, 2015). The central objects are the shifted products

$\sinh t$4

initially under conditions such as $\sinh t$5 and $\sinh t$6, with meromorphic continuation in the parameters afterward. The derivation begins from Laplace- and Fourier-transform formulas for $\sinh t$7, combines them with the standard integral representation of $\sinh t$8, and identifies the result with a product of two $\sinh t$9-functions (Nasri, 2015).

The key structural step is a hyperbolic reformulation of the one-sided Laplace integral. After substitutions such as $$J_\nu^2(x)+Y_\nu^2(x)=\frac{8}{\pi^2}\int_0^\infty K_0(2x\sinh t)\cosh(2\nu t)\,dt.$$0 or $$J_\nu^2(x)+Y_\nu^2(x)=\frac{8}{\pi^2}\int_0^\infty K_0(2x\sinh t)\cosh(2\nu t)\,dt.$$1, factors of the form $$J_\nu^2(x)+Y_\nu^2(x)=\frac{8}{\pi^2}\int_0^\infty K_0(2x\sinh t)\cosh(2\nu t)\,dt.$$2 are converted into kernels involving $$J_\nu^2(x)+Y_\nu^2(x)=\frac{8}{\pi^2}\int_0^\infty K_0(2x\sinh t)\cosh(2\nu t)\,dt.$$3 or $$J_\nu^2(x)+Y_\nu^2(x)=\frac{8}{\pi^2}\int_0^\infty K_0(2x\sinh t)\cosh(2\nu t)\,dt.$$4. This produces Nicholson-type formulas for parabolic cylinder functions in which the product is expressed by a single integral over $$J_\nu^2(x)+Y_\nu^2(x)=\frac{8}{\pi^2}\int_0^\infty K_0(2x\sinh t)\cosh(2\nu t)\,dt.$$5 with hyperbolic dependence in the kernel. The paper emphasizes that these relations are analogues, for parabolic cylinder functions, of the classical Nicholson integrals familiar from Bessel theory (Nasri, 2015).

Several reductions are singled out. When $$J_\nu^2(x)+Y_\nu^2(x)=\frac{8}{\pi^2}\int_0^\infty K_0(2x\sinh t)\cosh(2\nu t)\,dt.$$6, the shifted product collapses to $$J_\nu^2(x)+Y_\nu^2(x)=\frac{8}{\pi^2}\int_0^\infty K_0(2x\sinh t)\cosh(2\nu t)\,dt.$$7, and $$J_\nu^2(x)+Y_\nu^2(x)=\frac{8}{\pi^2}\int_0^\infty K_0(2x\sinh t)\cosh(2\nu t)\,dt.$$8 becomes the cleanest analogue of a classical Nicholson square-product formula. When $$J_\nu^2(x)+Y_\nu^2(x)=\frac{8}{\pi^2}\int_0^\infty K_0(2x\sinh t)\cosh(2\nu t)\,dt.$$9, one has

$\mathcal H(x)=\int K(x,y)H(y)\,dy$0

so the Nicholson-type relations become identities for Hermite polynomials multiplied by Gaussians. For $\mathcal H(x)=\int K(x,y)H(y)\,dy$1 and related half-integer values, the formulas reduce to modified Bessel functions, explicitly linking the new parabolic-cylinder formulas back to the older Bessel and Macdonald theory (Nasri, 2015).

The same transform machinery also yields local series expansions. By inserting the power series for $\mathcal H(x)=\int K(x,y)H(y)\,dy$2 into the Laplace kernel and integrating term by term, the paper derives convergent expansions for $\mathcal H(x)=\int K(x,y)H(y)\,dy$3 with even and odd parts separated. The paper states that the integral form is often better for global analytic properties and asymptotic analysis, whereas the series form is convenient near $\mathcal H(x)=\int K(x,y)H(y)\,dy$4 and for symbolic manipulation. It also notes numerical advantages: a single absolutely convergent integral may be more stable than evaluating two parabolic cylinder functions separately and multiplying them in regimes with cancellation (Nasri, 2015).

4. Jacobi, Ferrers, Gegenbauer, and Laguerre generalizations

A further generalization appears in "Integral representation for a product of two Jacobi functions of the second kind" (Cohl et al., 11 Aug 2025). There Nicholson-type relations are produced from a corrected kernel representation for

$\mathcal H(x)=\int K(x,y)H(y)\,dy$5

by taking opposite boundary values across the cut: $\mathcal H(x)=\int K(x,y)H(y)\,dy$6 The decisive boundary-value identity is

$\mathcal H(x)=\int K(x,y)H(y)\,dy$7

which the paper presents as the Jacobi analogue of

$\mathcal H(x)=\int K(x,y)H(y)\,dy$8

On that basis, Nicholson-type formulas become integral representations for sums of squares of first- and second-kind solutions on the cut, derived from off-cut product formulas for second-kind solutions (Cohl et al., 11 Aug 2025).

The central Jacobi Nicholson formula expresses

$\mathcal H(x)=\int K(x,y)H(y)\,dy$9

as a kernel integral over $(M_x-M_y)G=0$0 on $(M_x-M_y)G=0$1, with the kernel written either in terms of a parabolic-cylinder-type $(M_x-M_y)G=0$2-function or an associated Legendre $(M_x-M_y)G=0$3-function. The paper stresses that this is a kernel-form identity inherited from the product formula, not a stand-alone ad hoc representation. Specializations then yield Nicholson-type formulas for Ferrers functions, the classical Legendre relation

$(M_x-M_y)G=0$4

and corresponding Gegenbauer formulas (Cohl et al., 11 Aug 2025).

The same paper derives a confluent limit to Laguerre functions. Writing $(M_x-M_y)G=0$5 for the Laguerre function of the second kind off the cut and $(M_x-M_y)G=0$6 for the corresponding on-the-cut functions, it proves a product formula for $(M_x-M_y)G=0$7 and then specializes to $(M_x-M_y)G=0$8, $(M_x-M_y)G=0$9. The Nicholson-type Laguerre relation becomes

$\omega$0

with the right-hand side represented by a single integral involving $\omega$1, an exponential factor, and $\omega$2. The paper remarks that the apparent exponential growth from the Bessel kernel is canceled by the large-argument behavior of $\omega$3, ensuring convergence under the stated parameter restrictions (Cohl et al., 11 Aug 2025).

This line of work shifts the emphasis from square products of one function family to sums of squares of cut solutions. A plausible implication is that Nicholson-type relations can be understood as boundary-value identities for second-kind solutions as much as as hyperbolic product formulas, and that the passage between hyperbolic and trigonometric regimes is central rather than incidental (Cohl et al., 11 Aug 2025).

5. Heun-family kernel relations and transformation theory

The Heun literature develops Nicholson-type relations in the broader kernel-transform sense. In "Transformations of Heun's equation and its integral relations" (El-Jaick et al., 2010), the starting point is the Heun equation together with an integral transform

$\omega$4

where $\omega$5 is a Heun solution. The transform again solves the same equation provided the kernel satisfies

$\omega$6

and the bilinear concomitant

$\omega$7

takes equal values at the endpoints. The paper’s main contribution is to show that the transformation theory of Heun’s equation induces a transformation theory for kernels, generating new families from known ones by index and Möbius transformations (El-Jaick et al., 2010).

Two kernel classes are central there. Lambe–Ward-type kernels are given by a single hypergeometric function $\omega$8, while Erdélyi-type kernels have the separated form

$\omega$9

with $$\int_{0}^{\infty}\cos\!\left[\frac{3}{2}\xi\!\left(x+\frac{x^{3}}{3}\right)\right]\,dx$$0 and $$\int_{0}^{\infty}\cos\!\left[\frac{3}{2}\xi\!\left(x+\frac{x^{3}}{3}\right)\right]\,dx$$1 themselves hypergeometric. The paper then carries these kernels through confluence to the confluent Heun equation, producing kernels involving one confluent hypergeometric function, one Gauss hypergeometric function, products of two confluent hypergeometric functions, and mixed confluent/Gauss products. It explicitly suggests that this framework connects Heun, confluent Heun, double-confluent Heun, and Mathieu-type kernels within a common transformation theory (El-Jaick et al., 2010).

"Integral relations for solutions of confluent Heun equations" develops this program further for the CHE, RCHE, DHE, and RDHE (El-Jaick et al., 2013). For the confluent Heun equation it formulates the general integral relation

$$\int_{0}^{\infty}\cos\!\left[\frac{3}{2}\xi\!\left(x+\frac{x^{3}}{3}\right)\right]\,dx$$2

with kernel equation $$\int_{0}^{\infty}\cos\!\left[\frac{3}{2}\xi\!\left(x+\frac{x^{3}}{3}\right)\right]\,dx$$3 and a bilinear concomitant whose endpoint equality ensures that $$\int_{0}^{\infty}\cos\!\left[\frac{3}{2}\xi\!\left(x+\frac{x^{3}}{3}\right)\right]\,dx$$4 is again a solution. It then constructs several explicit kernel families: products of two confluent hypergeometric functions, hypergeometric $$\int_{0}^{\infty}\cos\!\left[\frac{3}{2}\xi\!\left(x+\frac{x^{3}}{3}\right)\right]\,dx$$5 confluent-hypergeometric kernels, kernels depending only on $$\int_{0}^{\infty}\cos\!\left[\frac{3}{2}\xi\!\left(x+\frac{x^{3}}{3}\right)\right]\,dx$$6, and hypergeometric kernels obtained by special choices of the separation constant (El-Jaick et al., 2013).

The most Nicholson-like consequences are explicit solution-to-solution transforms. The paper gives an integral relation sending a Jaffé power-series solution to a Leaver series in irregular confluent hypergeometric functions, another sending a Baber–Hassé power series to a regular $$\int_{0}^{\infty}\cos\!\left[\frac{3}{2}\xi\!\left(x+\frac{x^{3}}{3}\right)\right]\,dx$$7-series, and an integral transformation that produces a new spheroidal-wave solution as a $$\int_{0}^{\infty}\cos\!\left[\frac{3}{2}\xi\!\left(x+\frac{x^{3}}{3}\right)\right]\,dx$$8-series. In Appendix C it specializes the reduced CHE to Mathieu by taking

$$\int_{0}^{\infty}\cos\!\left[\frac{3}{2}\xi\!\left(x+\frac{x^{3}}{3}\right)\right]\,dx$$9

and states that the resulting kernels are equivalent to known Mathieu kernels in McLachlan. It also derives a transform from an RCHE power series to a Bessel-series solution. In this sense, Heun-family integral relations generalize the Nicholson idea from special-function products to systematic kernel intertwiners between local and global solution bases (El-Jaick et al., 2013).

6. Hilbert-transform variants and semi-axis Cauchy-kernel relations

Nicholson-type theory also has a neighboring branch based on singular-integral transforms rather than product formulas. "Integral relations associated with the semi-infinite Hilbert transform and applications to singular integral equations" does not explicitly use the term “Nicholson formula,” but the paper identifies its Bessel identities as structurally closest to Nicholson-type theory (Antipov et al., 2018). The central formula is

$\frac12\xi x^3=\omega^3$0

valid under $\frac12\xi x^3=\omega^3$1 with a specified branch convention. This expresses a semi-infinite Cauchy transform of one Bessel solution as a linear combination of two linearly independent Bessel solutions (Antipov et al., 2018).

Boundary-value versions on $\frac12\xi x^3=\omega^3$2 and $\frac12\xi x^3=\omega^3$3 lead to formulas involving $\frac12\xi x^3=\omega^3$4 and modified Bessel functions $\frac12\xi x^3=\omega^3$5. The paper derives these relations as limits of identities for Jacobi functions of the second kind $\frac12\xi x^3=\omega^3$6, using large-degree Jacobi-to-Bessel scaling near $\frac12\xi x^3=\omega^3$7. This use of a second-kind prelimit object is strongly reminiscent of the Jacobi Nicholson formulas derived from cut boundary values of $\frac12\xi x^3=\omega^3$8-functions, although here the final identities are Cauchy-kernel transforms rather than hyperbolic product integrals (Antipov et al., 2018).

The analytic role is concrete. The Bessel Hilbert-transform identity is used directly in a contact-mechanics model to convert a singular integral equation into an equation that can be solved by Hankel inversion, yielding a representation free of singular integrals. Elsewhere in the same paper, related Laguerre and confluent-hypergeometric Cauchy-kernel relations produce complete orthogonal systems of transformed functions and quadrature formulas for the Cauchy integral on a semi-axis. This suggests that Nicholson-adjacent relations are not limited to product evaluation; they also organize singular integral equations, orthogonal systems, and semi-axis transform theory (Antipov et al., 2018).

7. Terminological extensions, analytic uses, and disambiguation

Across the special-function literature, Nicholson-type relations serve several recurring purposes. They provide analytic continuation in parameters by isolating parameter dependence in gamma factors and a single kernel; they are well suited to asymptotic analysis because kernels such as $\frac12\xi x^3=\omega^3$9 have rapid decay and manageable saddle structure; and they can improve numerical stability by replacing a product of separately computed special functions with one absolutely convergent integral (Nasri, 2015). In Jacobi and Laguerre settings they also expose how off-cut second-kind products become on-cut sums of squares, making boundary behavior and branch structure part of the main theorem rather than a secondary technicality (Cohl et al., 11 Aug 2025).

At the same time, the label is not uniform across disciplines. In delayed differential equations, “Nicholson-type” refers to equations with Nicholson birth terms such as $$A(\rho)=\int_{0}^{\infty}\cos(\omega^{3}+\rho\omega)\,d\omega, \qquad \rho=3\left(\frac{\xi}{2}\right)^{2/3}.$$0. The periodic two-patch system with nonlinear density-dependent mortality studied in (Ossandón et al., 2020) uses the one-period identity

$$A(\rho)=\int_{0}^{\infty}\cos(\omega^{3}+\rho\omega)\,d\omega, \qquad \rho=3\left(\frac{\xi}{2}\right)^{2/3}.$$1

for periodic solutions and the averaged finite-dimensional map

$$A(\rho)=\int_{0}^{\infty}\cos(\omega^{3}+\rho\omega)\,d\omega, \qquad \rho=3\left(\frac{\xi}{2}\right)^{2/3}.$$2

as the core of a degree-theoretic existence proof. The paper explicitly states that it does not derive a variation-of-constants formula, Green-function representation, periodic integral equation, or Hammerstein fixed-point operator. Thus its integral relations are period-averaged balance relations, not classical Nicholson product formulas (Ossandón et al., 2020).

The most useful encyclopedic distinction is therefore between two families of usage. In special-function analysis, Nicholson-type integral relations are product, boundary-value, or kernel identities associated with second-order ODEs and special-function hierarchies. In mathematical biology, Nicholson-type systems are delay equations named for Nicholson’s blowflies, and any associated integral relations concern averaged balances or existence theory rather than classical hyperbolic kernel representations. The shared terminology records a historical name, but the mathematical content is substantially different (Nasri, 2015, Ossandón et al., 2020).