The Fourier coefficients of a Maass form phi for SL(n, Z) are complex numbers A(phi)(M), where M = (m(1), m(2), ..., m(n-1)) and m(1), m(2), ..., m(n-1) are non-zero integers. It is well known that coefficients of the form A(phi)(m(1), 1, ..., 1) are eigenvalues of the Hecke algebra and are multiplicative. We prove that the more general Fourier coefficients A(phi)(m(1), ..., m(n-1)) are also eigenvalues of the Hecke algebra and satisfy the multiplicativity relations A(phi)(m(1)m(1)', m(2)m(2)', ..., m(n-1)m(n-1)') = A(phi)(m(1), m(2), ..., m(n-1)).A(phi)(m(1)', m(2)', ..., m(n-1)') provided the products & prod;(n-1)(i=1) m(i) and & prod;(n-1)(i=1) m(i)' are relatively prime to each other.

Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on GL(1)) was used by Dirichlet to prove infinitely many primes in arithmetic progressions. Asymptotic orthogonality relations for GL$(n)$, with $n\le 3$, and applications to number theory, have been considered by various researchers over the last 45 years. Recently, the authors of the present work have derived an explicit asymptotic orthogonality relation, with a power savings error term, for GL$(4,\mathbb R)$. Here we we extend those results to GL$(n,\mathbb R)$ $(n\ge2)$. For $n\le 5$ our results are unconditional. In particular, the case $n=5$ represents a new result. The key new ingredient for the proof of the case $n=5$ is the theorem of Kim-Shahidi that functorial products of cusp forms on GL(2)$\times$GL(3) are automorphic on GL(6). For $n>5$ our results are conditional on two conjectures, both of which have been verified in various special cases. The first of these conjectures regards lower bounds for Rankin-Selberg L-functions, and the second concerns recurrence relations for Mellin transforms of GL$(n,\mathbb R)$ Whittaker functions. Our methods assume the Ramanujan conjecture at the infinite place for Maass cusp forms, but this assumption can be removed with a weakening in our error term. Central to our proof is an application of the Kuznetsov Trace formula, and a detailed analysis, utilizing a number of novel techniques, of the various entities -- Hecke-Maass cusp forms, Langlands Eisenstein series, spherical principal series Whittaker functions and their Mellin transforms, and so on -- that arise in this application.

Fourier coefficients of Eisenstein series figure prominently in the study of automorphic L-functions via the Langlands-Shahidi method, and in various other aspects of the theory of automorphic forms and representations. In this paper, we define Langlands Eisenstein series for SL(n, Z) in an elementary manner, and then determine the first Fourier coefficient of these series in a very explicit form. Our proofs and derivations are short and simple, and use the Borel Eisenstein series as a template to determine the first Fourier coefficient of other Langlands Eisenstein series.

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In this paper, we present a very simple explicit description of Langlands Eisenstein series for SL(n, ℤ). The functional equations of these Eisenstein series are heuristically derived from the functional equations of certain divisor sums and certain Whittaker functions that appear in the Fourier coefficients of the Eisenstein series. We conjecture that the functional equations are unique up to a real affine transformation of the s variables defining the Eisenstein series and prove the uniqueness conjecture in certain cases.

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Using a recursive formula for the Mellin transform T-n,T-a(s) of a spherical, principal series GL(n, R) Whittaker function, we develop an explicit recurrence relation for this Mellin transform. This relation, for any n >= 2, expresses T-n,T-a(s) in terms of a number of "shifted" transforms T-n,T-a(s + Sigma), with each coordinate of Sigma being a non-negative integer. We then focus on the case n = 4. In this case, we use the relation referenced above to derive further relations, each of which involves "strictly positive shifts" in one of the coordinates of s. More specifically: each of our new relations expresses T-4,T-a(s) in terms of T-4,T-a(s+Sigma) and T-4,T- a (s+Omega), where for some 1 <= k <= 3, the kth coordinates of both Sigma and Omega are strictly positive. Next, we deduce a recurrence relation for T-4,T-a(s) involving strictly positive shifts in all three s(k) 's at once. (That is, the condition "for some 1 <= k <= 3" above becomes "for all 1 <= k <= 3.") These additional relations on GL(4, R) may be applied to the explicit understanding of certain poles and residues of T-4,T-a(s). This residue information is, as we describe below, in turn relevant to recent results concerning orthogonality of Fourier coefficients of SL(4, Z) Maass forms, and the GL(4) Kuznetsov formula. (C) 2020 Elsevier Inc. All rights reserved.

Abstract Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on $\mathrm {GL}(1)$) was used by Dirichlet to prove infinitely many primes in arithmetic progressions. Orthogonality relations for $\mathrm {GL}(2)$ and $\mathrm {GL}(3)$ have been worked on by many researchers with a broad range of applications to number theory. We present here, for the first time, very explicit orthogonality relations for the real group $\mathrm {GL}(4, \mathbb R)$ with a power savings error term. The proof requires novel techniques in the computation of the geometric side of the Kuznetsov trace formula.

In this paper, we use combinatorial group theory and a limiting process to connect various types of hypergeometric series, and of relations among such series. We begin with a set S of 56 distinct translates of a certain function M , which takes the form of a Barnes integral, and is expressible as a sum of two very-well-poised _9F_8 hypergeometric series of unit argument. We consider a known, transitive action of the Coxeter group W(E_7) on this set. We show that, by removing from W(E_7) a particular generator, we obtain a subgroup that is isomorphic to W(D_6) , and that acts intransitively on S , partitioning it into three orbits, of sizes 32, 12, and 12, respectively. Taking certain limits of the M functions in the first orbit yields a set of 32 J functions, each of which is a sum of two Saalschützian _4F_3 hypergeometric series of unit argument. The original action of W(D_6) on the M functions in this orbit is then seen to correspond to a known action of this group on this set of J functions. In a similar way, the image of each of the size-12 orbits, under a similar limiting process, is a set of 12 L functions that have been investigated in earlier works. In fact, these two image sets are the same. The limiting process is seen to preserve distance, except on pairs consisting of one M function from each size-12 orbit. Finally, each known three-term relation among the J and L functions is seen to be obtainable as a limit of a known three-term relation among the M functions.

As mathematics teachers, we hope our students will approach problems with a spirit of creativity. One way to both model and encourage this spirit - and, at the same time, to keep ourselves from getting bored - is through creative approaches to problem design. In this paper, we discuss TACTivities, mathematical activities with a tactile component, as a creative outlet for those of us who teach mathematics, and as a resource for stimulating creative thinking in our students. We use examples, such as our derivative fridge magnets TACTivity, to illustrate the main ideas. We emphasize that TACTivities can be engaging, to teachers and learners alike, at any level of mathematics, by including examples from different mathematics courses (calculus and mathematics for elementary teachers). As an example, our derivative fridge magnets have moving pieces of words that look like small refrigerator magnets. These small pieces can be combined to make true mathematical statements, of the form d/dx (some function) = some other function. There was creativity involved in the creation of these magnets, as the mathematics had to be challenging enough not to bore students yet have an easy entry for students to be successful. The students working with the magnets can use their creativity along with their mathematical knowledge while learning and/or reviewing a mathematical concept-in this case derivatives. We will expand on the creative side of the creation and implementation of TACTivities in this paper. Note that our definition of tactile only means moving pieces (usually pieces of paper), as this is different than work from others that involves tactile props such as pipe cleaners, yarn, Spirographs, building blocks, and so on. This other work is invaluable, and we use props like these ourselves at times, but we believe that our TACTivities add a different dimension to tactile learning.

Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on GL(1)) was used by Dirichlet to prove infinitely many primes in arithmetic progressions. Orthogonality relations for GL(2) and GL(3) have been worked on by many researchers with a broad range of applications to number theory. We present here, for the first time, very explicit orthogonality relations for the real group GL(4,ℝ) with a power savings error term. The proof requires novel techniques in the computation of the geometric side of the Kuznetsov trace formula. An appendix by Bingrong Huang gives new bounds for the relevant Kloosterman sums.

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We investigate sums \(J(\vec {x})\) and \(L(\vec {x})\) of pairs of normalized Saalschützian \({}_4F_3(1)\) hypergeometric series, and develop a theory of relations among these J and L functions. The function \(L(\vec {x})\) has been studied extensively in the literature and has been shown to satisfy a number of two-term and three-term relations with respect to the variable \(\vec {x}\). More recent works have framed these relations in terms of Coxeter group actions on \(\vec {x}\) and have developed a similar theory of two-term and three-term relations for \(J(\vec {x})\). In this article, we derive “mixed” three-term relations, wherein any one of the L (respectively, J) functions arising in the above context may be expressed as a linear combination of two of the above J (respectively, L) functions. We show that, under the appropriate Coxeter group action, the resulting set of three-term relations (mixed and otherwise) among J and L functions partitions into eighteen orbits. We provide an explicit example of a relation from each orbit. We further classify the eighteen orbits into five types, with each type uniquely determined by the distances (under a certain natural metric) between the J and L functions in the relation. We show that the type of a relation dictates the complexity (in terms of both number of summands and number of factors in each summand) of the coefficients of the J and L functions therein.

This article presents several of the challenges facing postsecondary mathematics education and describes how the undergraduate Learning Assistant (LA) program has been used as a catalyst to engage faculty and students in redesigning opportunities to learn mathematics. Characteristics of the LA program that have been used to transform introductory undergraduate science courses are discussed. We then describe how the LA program was implemented in a mathematics department vis-à-vis the specific contextual features of a mathematics department at the University of Colorado Boulder.