Zonal polynomials and a multidimensional quantum Bessel process

Abstract

Introduction: Primary cutaneous gamma-delta T cell lymphoma (CGD-TCL) is a rare form of cutaneous T cell lymphoma (CTCL) associated with a poor prognosis and recently categorized with peripheral T cell lymphomas of the skin.Case presentation: A 65-year-old Caucasian female presented for evaluation of ''nodules'' on her lower extremities. In 2012, she developed tender, pruritic nodules in her right calf with additional lesions appearing over the next several months. On examination there were multiple erythematous subcutaneous nodules on her bilateral lower extremities, buttocks, and right upper arm.Staging: Multiple biopsies from lesions showed an atypical T cell lymphoid infiltrate with prominent granulomatous inflammation involving the deep dermis. Most lymphocytes expressed CD3 with a CD4:CD8 ratio of 1:1 with a subpopulation also positive for CD56, TCR gamma, and TIA-1. The interpretation was CGD-TCL with prominent granulomatous inflammation. Bone marrow and nodes showed no evidence of lymphoma. Initial PET/CT scan revealed abnormal uptake in the subcutaneous tissue of the trunk and extremities. ANA and anti-ds DNA were negative.Treatment: The patient initially used clobetasol ointment but progressed. Her lesions were controlled on oral bexarotene 225 mg daily and prednisone 20 mg daily, which was decreased to 10 mg over four months. In January 2014, hydroxychloroquine 200 mg BID was added. As she developed new lesions, confirmed with PET/CT imaging, her prednisone therapy was increased to 40 mg daily. Future treatment options include romidepsin, a CD3-diphtheria fusion protein; praletrexate, a folic acid inhibitor; total body electron beam radiation and allotransplant; or standard chemotherapy.Discussion: This case highlights a rare form of panniculitis-like cutaneous lymphoma, which represents \1% of all CTCL. Autoimmune diseases have been reported in 20% of patients with the alpha-beta phenotype. Our patient has the gamma-delta phenotype, which is associated with a very aggressive clinical course with a five year survival of \10%. Her course has been more indolent with partial response to bexarotene and prednisone. This case highlights new and experimental treatment options for peripheral T-cell lymphomas. Conclusion: Primary CGD-TCL is a rare form of peripheral T-cell lymphoma arising in the skin and may mimic inflammatory panniculitis. Prompt excisional biopsy and recognition is crucial in determining the disease course and treatment.Commercial support: None identified. 1660Severe leonine facies in a case of folliculotropic mycosis fungoides

Asymmetric Simple Exclusion Process with Open Boundaries and Quadratic Harnesses

“…[1]). Other examples of quadratic harnesses are quantum Bessel process -QH(θ, η, 0, 0, 1) (see [3], [10], or [34]), and a more general bi-Poisson process -QH(θ, η, 0, 0, q) (see [9]), or the q-Brownian motion -QH(0, 0, 0, 0, q) (see, e.g. [7]).…”

Hypergroups and Quantum Bessel Processes of Non-integer Dimensions

“…The rest of the proof consists of a careful analysis of (7.2) in all the cases leading to (2.6)-(2.10) (with δ instead of d), and is based on the same general approach laid out in [10]. But, for completeness, we outline it here (see also Remark 8.2).…”

“…Since the first coordinate of the quantum Bessel process always follows a uniform motion to the right, it makes sense to denote the whole process as (t, X t ) t . It can be shown in a way similar to Theorem 4.1 in [10] that the second coordinate (X t ) t after a slight rescaling becomes a particular case of quadratic harness (for the definition, see [4]), called classical bi-Poisson process, which was first introduced in [6] and presented as a member of a larger class of bi-Poisson processes in [5].…”

Infinitesimal generators for a class of polynomial processes

“…This quadratic harness appeared under the name quantum Bessel process in [4] (see also [20] for a multidimensional version), and as classical bi-Poisson process in [11]. Then equation (2.4) assumes the form (5.4) HF − FH = E + θH + η(F − tH) = E + ηF + (θ − tη)H.…”