https://forum.jquery.com/topic/download-a-file-via-ajax#14737000005999144

<!DOCTYPE html>
<html>

  <head>
    <script data-require="jquery@2.1.1" data-semver="2.1.1" src="//cdnjs.cloudflare.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
    <link rel="stylesheet" href="style.css" />
<script src="jquery.binarytransport.js"></script>
    <script src="script.js"></script>

  </head>

  <body>
    <button>download pdf</button>
  </body>

</html>

$(function() {
  $("button").click(function() {
    $.ajax({
      url: "http://cors.jejaju.com/Snickerdoodles.pdf",
      type: "GET",
      dataType: 'binary',
      success: function(result) {
        var url = URL.createObjectURL(result);
        var $a = $('<a />', {
          'href': url,
          'download': 'document.pdf',
          'text': "click"
        }).hide().appendTo("body")[0].click();
        setTimeout(function() {
          URL.revokeObjectURL(url);
        }, 10000);
      }
    });
  });
});

/* Styles go here */

Written for https://forum.jquery.com/topic/download-a-file-via-ajax#14737000005999144

 /**
 *
 * jquery.binarytransport.js
 *
 * @description. jQuery ajax transport for making binary data type requests.
 * @version 1.0 
 * @author Henry Algus <henryalgus@gmail.com>
 *
 */

// use this transport for "binary" data type
$.ajaxTransport("+binary", function(options, originalOptions, jqXHR){
    // check for conditions and support for blob / arraybuffer response type
    if (window.FormData && ((options.dataType && (options.dataType == 'binary')) || (options.data && ((window.ArrayBuffer && options.data instanceof ArrayBuffer) || (window.Blob && options.data instanceof Blob)))))
    {
        return {
            // create new XMLHttpRequest
            send: function(_, callback){
		// setup all variables
                var xhr = new XMLHttpRequest(),
                    url = options.url,
                    type = options.type,
		// blob or arraybuffer. Default is blob
                    dataType = options.responseType || "blob",
                    data = options.data || null;
				
                xhr.addEventListener('load', function(){
                    var data = {};
                    data[options.dataType] = xhr.response;
		// make callback and send data
                    callback(xhr.status, xhr.statusText, data, xhr.getAllResponseHeaders());
                });

                xhr.open(type, url, true);
                xhr.responseType = dataType;
                xhr.send(data);
            },
            abort: function(){
                jqXHR.abort();
            }
        };
    }
});